Nucleation and Droplet Growth During Co-condensation of Nonane and D2O in a Supersonic Nozzle
DISSERTATION
Presented in Partial Fulfillment of the Requirements for the Degree Doctor of Philosophy in the Graduate School of The Ohio State University
By
Harshad Narayan Pathak
Graduate Program in Chemical and Biomolecular Engineering
The Ohio State University
2013
Dissertation Committee:
Barbara E. Wyslouzil, Advisor
Isamu Kusaka
Bhavik Bakshi Copyright by
Harshad Narayan Pathak
2013 Abstract
Raw natural gas consists mainly of methane and has impurities like water vapor, higher alkanes, H2S etc. Dehydration of natural gas is important to prevent hydrate formation in pipelines carrying natural gas over long distances. Traditionally, dehydration is done using chemical methods like pressure swing absorption and glycol dehydration. An alternate method of dehydration is by using a mechanical process of supersonic separation. In this method, raw natural gas is cooled down by adiabatic expansion resulting in condensation of water vapor and higher alkanes.
The goal of this work is to understand the nucleation and droplet growth when droplet sizes are of the order of nm and timescales are of the order of microseconds when water and alkanes, two substances which are immiscible, condense together. We use supersonic nozzles in this work where cooling rates are of the order of 105-106 K/s. The supersonic velocities of the flow enable measurements on a resolution of the order of microseconds.
Pressure trace measurement (PTM) is our basic experimental technique and it characterizes the flow by measuring the pressure profile inside the supersonic nozzle as the vapor-gas mixture expands and vapor condenses inside the nozzle. These experiments give us the initial estimate of temperature, density, velocity and mass fraction of the condensate. We use Fourier transform infrared spectroscopy (FTIR) to get the composition of the condensed liquid/vapor. To determine the amount of nonane
ii condensed, we fit the measured spectrum of nonane to a linear combination of a well- characterized vapor and liquid spectrum. For D2O analysis, we calculate D2O vapor concentration by analyzing the vibrational-rotational spectrum of O-D stretch region. The size and number of droplets is characterized using small angle x-ray scattering (SAXS) that are performed in Argonne National Laboratory.
The nucleation rates for pure D2O and nonane agree with previous measurements done by other researchers. The subsequent process of growth of the droplets can be sensitive to droplet temperatures Td. For pure nonane droplets, we observe that Td is not important enough to alter the growth rates unlike pure D2O. The growth of D2O droplets is further affected by coagulation once condensation has slowed down. We also observe that when nonane and D2O both are condensing, the presence of nonane inhibits D2O condensation even when D2O dominates the nucleation process.
Prediction of the droplet structure of composite nonane-D2O droplets is challenging because the SAXS spectra of these droplets does not fit to standard shapes like spheres or core-shell structures. The small size of these droplets makes it possible to study them through molecular dynamics simulations. Our collaborators conduct simulations of these droplets and calculate the scattering behavior for those shapes. The SAXS spectra are fit to scattering from shapes derived from both density functional theory (DFT) calculations and molecular dynamics (MD) simulations. Although the ‘lens-on-sphere’ structures derived from MD simulations fits the scattering spectra better than all other structures which we tested, the overall composition from this structure predicts that the amount of
D2O condensed is 30-40% less than that measured from FTIR. iii Dedication
Dedicated to my mother, father and all my close friends and family.
iv Acknowledgments
I would like to thank my research advisor, Dr. Barbara Wyslouzil for her mentorship. She is the best advisor one could ask for. I am thankful to her for tolerating my odd working hours. Research discussions with her have been immensely helpful in developing my understanding of concepts and advancing my research forward. She has been a true source of knowledge and support.
I would also like to thank my collaborators Dr. Judith Wölk and Dr. Gerald Wilemski.
Listening to their research ideas and working on their input has always helped me learn something new. Another source of guidance in my first year was Dr. Shinobu Tanimura whose meticulousness and self-discipline has been an inspiration.
I would also like to thank National Science Foundation, Argonne National Lab, the
American Chemical Society’s Petroleum Research Fund and Deutsche Akademischer
Austausch Dienst (German Academic Exchange Service) for funding this work. I am also thankful to instrument scientist, Dr. Soenke Seifert from Argonne National Lab for his continuous help during our beamtime at Argonne National Lab. I would like to acknowledge the guidance which I received during my first few months from Dr.
Hartawan Laksmono, Dr. Kelley Mullick and Dr. Ashutosh Bhabhe who helped me ease into graduate work. I am also thankful to Dirk Bergmann, Daniel Weckstein and Dr.
v Alexandra Manka for helping me perform experiments. I would like to thank my friendly and trustworthy colleagues like Viraj Modak, Dr. Anthony Duong, Matthew Gallovic,
Alyssa Robson, Gauri Nabar and Andrew Amaya who have made my workplace enjoyable. I am also thankful to Matthew Souva who helped me reinstall important software on my computer when my hard disk crashed two months before the dissertation defense.
I am also grateful to all my friends in Columbus whose company has never made me feel lonely. Special thanks to Dr. Nihar Phalak, Prateik Singh, Anshuman Fuller and
Somsundaram Chettiar who have been my roommates and supported me through thick and thin. I also consider myself fortunate to have been friends with Kalpesh Mahajan,
Hrishikesh Munj, Niranjani Deshpande, Mandar Kathe, Dr. Shreyas Rao, Dr. Preshit
Gawade, Dr. Shweta Singh and Dr. Kartik Ramasubramanian. I would also like to thank
Dr. Bryan Mark and his family who have been kind enough to invite me to their
Thanksgiving dinner every year.
I would like to thank my parents, Narayan Pathak and Anuradha Pathak, for their love and support. They have always encouraged me to be independent and follow my dreams which at times have been unconventional. They have helped me shape my self- confidence and will power and for that, I am forever grateful.
vi Vita
August 2004- June 2008 ...... Bachelor of Chemical Engineering,
Institute of Chemical Technology,
Mumbai, India
September 2008- August 2009...... University Fellow,
The Ohio State University
September 2008- June 2011...... M.S. in Chemical Engineering,
The Ohio State University
September 2009- August 2010...... Graduate Teaching Associate,
First Year Engineering Program,
The Ohio State University
September 2009- present...... Graduate Research Associate,
The Ohio State University
vii Publications
1. H. Pathak, K. Mullick, S. Tanimura, and B. E. Wyslouzil. Nonisothermal Droplet
Growth in the Free Molecular Regime. Aerosol Science and Technology 47, 1310-
1324 (2013).
2. Bhabhe, H. Pathak, and B. E. Wyslouzil. Freezing of heavy water (D2O)
nanodroplets, Journal of Phys. Chem. A. 117, 5472-5482 (2013).
3. V. Modak; H. Pathak; M. Thayer; S.J. Singer and B. E. Wyslouzil. Experimental
evidence for surface freezing in supercooled n-alkane nanodroplets. Physical
Chemistry Chemical Physics 15, 6783-6795 (2013).
4. A. Manka, H. Pathak, S. Tanimura, J. Wölk, R. Strey, and B. E. Wyslouzil.
Freezing water in no-man’s land. Physical Chemistry Chemical Physics 14, 4505-
4516 (2012).
Fields of Study
Major Field: Chemical and Biomolecular Engineering
viii Table of Contents
Abstract...... ii
Dedication...... iv
Acknowledgments...... v
Vita...... vii
Chapter 1 Introduction ...... 1
1.1 Phase Transitions ...... 2
1.2 Classical Nucleation Theory...... 3
1.3 Motivation for the current work...... 6
1.4 Objective and Thesis Outline...... 7
Chapter 2 Experimental Methods ...... 12
2.1 Materials ...... 13
2.2 Experimental set-up ...... 13
2.3 Experimental techniques...... 16
2.3.1 Static Pressure Measurements (PTM)...... 16
2.3.2 Fourier Transform Infrared Spectroscopy (FTIR) ...... 17
ix 2.3.3 Small Angle X-ray Scattering (SAXS)...... 22
2.3.4 Integrated data analysis...... 25
Chapter 3 The Growth of Nonane and D2O Nanodroplets ...... 30
3.1 Introduction...... 31
3.2 Droplet Growth Laws: ...... 33
3.3 Droplet Growth Experiments...... 37
3.3.1 Nonane ...... 39
3.3.2 D2O ...... 53
3.4 Summary and Conclusions ...... 62
Chapter 4 Co-Condensation of Nonane and D2O in a Supersonic Nozzle ...... 73
4.1 Introduction...... 74
4.2 Experiments and their analysis ...... 77
4.2.1 Unary Condensation and Nucleation ...... 79
4.2.2 Binary Condensation and Nucleation ...... 88
4.2.3 Effect of Nonane on D2O nucleation ...... 97
4.3 Summary and Conclusions ...... 108
Chapter 5 The Structure of D2O-Nonane Nanodroplets ...... 114
5.1 Introduction...... 115
5.2 Theory...... 116
5.3 Experiments and their analysis ...... 119
x 5.3.1 Analysis of SAXS spectra for unary droplets...... 120
5.3.2 Analysis of Binary SAXS spectra...... 123
5.4 Summary and Conclusions ...... 139
Chapter 6 Conclusions and Future Work...... 143
Bibliography ...... 150
Appendix A: Appendix to Chapter 2: Fortran Code to Analyze Pressure Trace Measurements ...... 171
A.1 FORTRAN code used to calculate the temperature T, density ρ, mass fraction
condensed g and velocity u using p and (A/A*)dry as input...... 172
A.2 FORTRAN code used to calculate the experimental parameters using p and gfit from SAXS as input...... 204
A.3: FORTRAN code used to calculate the experimental parameters using p and gfit of
nonane and D2O from FTIR as input ...... 239
Appendix B Thermo-physical Properties of D2O, Nonane and Nitrogen...... 275
Appendix C:Appendix to Chapter 3:Droplet Temperatures and Growth Rate Calculations for Pure Nonane and Pure D2O Nanodroplets ...... 283
Appendix D: Appendix to Chapter 4: The Experimental Parameters of Binary Traces. 287
Appendix E: Appendix to Chapter 5: Analytical expressions of the Form factor and Guinier analysis for the Lens-on-Sphere Structure...... 297
E.1 Form factor for the lens-on-sphere structure ...... 298
E.2 Guinier analysis for the lens-on sphere structure ...... 300
xi List of Tables
Table 1 It shows the values of n* and ∆G* for different values of S when water vapor is condensing at 298 K. We can see that as S increases, n* and ∆G* decrease...... 5
Table 2 A summary of the position resolved condensation experiments where p0, T0 and pv0 are the pressure, temperature and partial pressure of condensable vapor at the inlet to the nozzle. The carrier gas in all experiments was N2...... 39
Table 3 All experiments used nitrogen as the carrier gas. TJmax and gmax values are given in this table. The values of pv0 are nominal values and the actual values are within 20 Pa of these values and are mentioned in Appendix D...... 79
Table 4 gmax and TJmax as a function of the nominal pv0 for D2O and nonane. The actual pv0 are within 20 Pa of these values and the exact value corresponding to each experiment is listed in Table 6. TJmax is the temperature corresponding to the maximum nucleation rate and gmax is the mass fraction of the condensible vapor entering the nozzle...... 120
Table 5 The sensitivity of fit parameters to different droplet structures. The numbers are bolded when mass balance is violated...... 128
Table 6 N and rG derived using different analyses of SAXS spectra at nozzle exit for the nine case of binary condensation...... 136
Table 7 g/gmax for nonane and D2O at the nozzle exit along with the weight percent of
D2O for the nine sets of experiments. FTIR results for pv0 = 618 Pa nonane and 367 Pa
D2O shows very little condensation of D2O which is difficult to quantify. This is the case where nucleation and the subsequent droplet growth is dominated by nonane...... 138
xii Table B 1 Thermophysical properties of D2O...... 276
Table B 2 Thermo-physical properties of nonane...... 278
Table B 3 Thermo-physical properties of N2...... 280
Table D 1 The exact pv0 of nonane and D2O for different experiments where nonane and
D2O vapors condense together with their mass fractions gmax which enter the nozzle. ~ NSAXS is the maximum number density of the aerosol measured by SAXS before coagulation, ΔtJmax is the characteristic nucleation time and Jmax are the measured nucleation rates. SD2O and Snonane are the saturation ratios reached by nonane and D2O respectively at Jmax...... 295
xiii List of Figures
Figure 1: ∆G as a function of n for different values of S calculated from Equation (1.2). We can see that as the value of S increases, n* and ∆G* decrease demonstrating that the barrier to nucleation is lowered when S is high...... 5
Figure 2 Experimental set-up to generate aerosols of D2O, nonane and their mixtures in a supersonic nozzle...... 16
Figure 3 A schematic of the arrangement of FTIR Spectrometer to measure the absorption spectrum of mixture inside the nozzle...... 20
Figure 4 (a)The molar absorptivity D2O,v of D2O vapor calculated by measuring several spectra prior to condensation. The temperature and the partial pressure of D2O are given in the legend. (b) The comparison of mole fraction of D2O vapor between the FTIR and
TDLAS for two different pv0 of D2O...... 22
Figure 5 Nonane droplet growth experiments for T0 = 35 C and pv0 = 625 Pa. (a) The measured pressures and estimated temperatures of the expanding supersonic flow. The temperature estimates are based on the integrated analysis described in Sec II.D that incorporates both the PTM and the SAXS data. (b) The average droplet radius and the spread of the droplet size distributions. The solid line in (b) is a 3 parameter sigmoid fit to the average radii (c) The mass fraction of the condensate g based on SAXS, gSAXS approaches the total mass fraction of condensible, gmax, near the nozzle exit. The supersaturation of the nonane vapor peaks near onset. (d) The normalized nucleation rate based on the classical nucleation theory illustrates that particle formation is localized in ~ the nozzle. The predicted specific number densities N are calculated by integrating the nucleation rate expression with respect to time, and then scaling these values to match the ~ experimental number densities at the nozzle exit. In the absence of coagulation N is conserved. The specific number densities measured using SAXS follow the trend
xiv predicted by theory. The error bars in (c) and (d) represent the systematic uncertainty of ±5% that arises from the estimated uncertainty in the absolute calibration procedure for the SAXS experiments...... 40
Figure 6 Nonane droplet temperatures and growth rates for experiments conducted at T0 =
35 C and pv0 =625 Pa are calculated as described in the text. The legend for (a) and (b) are the same. The error bars on Td,exp are based on a 5% uncertainty in calculating the number density and a 5% uncertainty in calculating the condensation rate, dg/dt based on the subjectivity of the sigmoidal fit. (a) The experimental and theoretical implicit droplet temperatures. (b) The temperature difference between the droplets and the surrounding gas mixture. (c) The experimental and theoretical growth rates agree well even when Td is much higher than T. The isothermal and non-isothermal growth laws predict essentially the same growth rate...... 49
Figure 7 D2O droplet growth experiments for T0 = 35 C, pv0 = 683 Pa, and p0 = 30.2 kPa. (a) The measured pressure and estimated temperatures of the expanding supersonic flow. (b) The average droplet radii and the spread of the droplet size distribution. (c) The mass fraction of condensate g as determined by SAXS and PTM, and a fit to the combined data. The water vapor supersaturation peaks near onset. (d) The predicted normalized nucleation rates and specific number densities are compared to the measured specific number densities. The predicted number densities are scaled to match the maximum in the observed number densities. The decrease in number density is due to coagulation and coincides with a slow increase in particle size. The error bars in (c) and (d) represent the systematic uncertainty of ±5% that arises from uncertainty in the absolute calibration procedure for the SAXS experiments...... 55
Figure 8 The droplet temperatures and growth rates for D2O experiments at T0 = 35 C, pv0
= 683 Pa, and p0 = 30.2 kPa are calculated assuming qc=qe=1. The legend for (a) and (b) are the same. The error bars on Td,exp are based on a 5% uncertainty in calculating the number density and a 5% uncertainty in calculating the condensation rate, dg/dt based on the subjectivity of the sigmoidal fit. (a) The experimental and theoretical implicit droplet
xv temperatures. (b) The temperature difference between the droplets and the surrounding gas mixture. (c) The experimental and theoretical growth rates. Using the values of Td,exp in Equation (3.1) yields negative growth rates suggesting the droplets are unstable...... 57
Figure 9 The droplet temperatures and growth rates for the same experimental conditions as Figures 3 and 4, but with qc=1 and qe=0.5 (a) Td,exact agrees better with Td,exp when qe=0.5. (b) The growth rates match the experimental values more closely when qe is reduced...... 59
Figure 10 (a) The non-isothermal growth rates predicted using Td,exact and that ignore or incorporate coagulation, are compared to the measured growth rates for the experiments with pv0=683 Pa D2O. Coagulation is only considered for t > ~20 µs because coagulation rates cannot be accurately measured at earlier times. (b) The non-isothermal growth rates predicted using Td,exact and that ignore or incorporate coagulation are compared to the measured growth rates for pv0=520 Pa D2O...... 62
Figure 11 The experimental measurements for pv0 of n-nonane equal to 322 Pa, 489 Pa and 625 Pa. (a) the pressure ratio and temperature for the mixture as it expands in the nozzle. The dashed line is the isentrope and the solid line is the measured pressure or temperature. (b) the average radii, r and the spread σ of the droplet size distribution. (c) the mass fraction, g of the liquid as it condenses in the nozzle. The dashed line is the maximum value gmax, the symbols, gSAXS are calculated from SAXS analysis and the lines which closely follow the symbols are calculated from the integrated data analysis, gfit. (d) The normalized nucleation rates on the left axis and the number densities on the right axis. The dashed lines are the number densities predicted by classical nucleation theory but scaled to match the number densities at the exit. The symbols are number densities ~ from the analysis of SAXS data N SAXS ...... 82
Figure 12 Experimental results for pv0 of D2O equal to 346 Pa, 520 Pa and 683 Pa. The black symbols correspond to the highest pv0, white symbols correspond to the lowest pv0 and the grey symbols correspond to the intermediate pv0. (a) the pressure ratio on the left
xvi axis and the temperature on the right axis. The dashed line represents the isentrope calculation. (b) average radii, r and the spread σ of the size distribution. (c) mass fraction, g of the condensate. The dashed line is the maximum value gmax, the symbols, gSAXS are calculated from SAXS analysis and the lines which closely follow the symbols are calculated from the integrated data analysis, gfit. (d) scaled nucleation rates on the left ~ ~ axis and the measured ( N SAXS ) and calculated ( N CNT ) number densities on the right axis. The calculated number densities are scaled to match the maximum number densities.... 85
Figure 13 The comparison of measured nucleation rates with previous work for a)
Nonane and b) D2O. Temperature at the maximum nucleation rate, TJmax is mentioned next to the symbols...... 87
Figure 14 The comparison of FTIR and the integrated analysis and PTM when (a) pv0=
625 Pa of nonane is condensing and (b) pv0= 683 Pa of D2O is condensing...... 88
Figure 15 A typical IR spectrum during co-condensation of D2O and nonane. The vapor peak for O-D stretch used to quantify the D2O vapor phase concentration is shaded in the figure; the values of pv0 are the stagnation pressures for the vapor...... 90
Figure 16 Experimental parameters when n-nonane (pv0= 487 Pa) and D2O (pv0= 538 Pa) both condense together and their comparison to cases where pure vapor condenses
(pv0=520 Pa for D2O and pv0=489 Pa for n-nonane). a) The temperature profile for the binary case (black line) and its comparison to the case where pure vapor condenses (grey lines). The isentrope is depicted by a dashed line. b) The average radii and the spread of the size distribution compared to the pure n-nonane and pure D2O droplets (grey symbols). c) mass fraction condensed as predicted from FTIR, gFTIR (black symbols for fitting using the vapor spectrum and white symbols for fitting using the liquid spectrum).
The black line which follows the symbols is gfit. The grey lines are gfit for cases where ~ pure components condense. The dashed-dot line is gmax. d) N SAXS and ~ ~ N scaledCNT compared with N SAXS for cases where pure components condense...... 96
xvii Figure 17 S/Sunary for D2O and nonane where D2O (pv0=704 Pa) and nonane (pv0=319 Pa) both are condensing and D2O dominates nucleation. TJmax is 216 K in this case...... 98
Figure 18 Condensation experiments for pv0=700 Pa D2O and varying amounts of nonane. (a) The nucleation pulses and characteristic times estimated based on CNT for pure D2O nucleation during unary condensation of D2O and during D2O-nonane co- condensation. (b) The measured number densities...... 100
Figure 19 (a) The nucleation rates of D2O as a function of the partial pressure of nonane pv at the conditions corresponding Jmax for D2O. The crosses show the predictions from
Feder et al’s non-isothermal nucleation theory. (b) The nucleation rate of D2O as a function of Snonane at the conditions corresponding to Jmax for D2O. The case when pv0=360 Pa D2O + 490 Pa nonane is not shown here because SD2O is reduced from 188 to 163 due to competitive nucleation...... 101
Figure 20 The experimental parameters when both nonane (pv0= 618 Pa) and D2O (pv0=
367 Pa) are condensing together compared to the conditions when pure nonane (pv0=625
Pa) and pure D2O (pv0=346 Pa) are condensing. The symbols are explained in Figure 16. This is the only case where nonane dominates nucleation and after the onset of nucleation, there is negligible condensation of D2O as seen from the FTIR result in figure c. In figure d, the nucleation pulse is calculated for nonane...... 105
Figure 21 S/Sunary for the case where nonane appears to dominates nucleation(TJmax =210
K). D2O (pv0 = 367 Pa) and nonane (pv0 = 618 Pa)...... 107
Figure 22 (a) An illustration of the sphere-in-sphere structure derived from DFT. There is a sphere of D2O (radius=R2) inside a sphere of nonane (radius=R1). d is the distance between the two centers. θc is the contact angle of nonane on D2O and is 0 in this case. (b) 0 An illustration of the lens-on-sphere structure derived from MD simulations. θc is 76 in this case. (c) An illustration of the sphere-outside-sphere structure derived from 0 simplifying the MD simulations results. θc is 180 in this case...... 118
xviii Figure 23 A typical fitting case where we fit the pure component spectra to theoretical scattering from polydisperse spheres. The value of r are obtained from the fitting parameters. (a) Nonane spectra are shown when droplets are at the nozzle exit. The spectra for r =25 nm and r =20 nm have been offset by 100 and 10 respectively for clarity. (b) D2O spectra are shown when droplets are at the nozzle exit. The spectra for r =5.1 nm and r =4.6 nm have been offset by 100 and 10 respectively for clarity. . 121
Figure 24 The reduced chi-squared values of SAXS fitting as we move downstream of the nozzle...... 123
Figure 25 (a) The fits of binary SAXS spectra assuming the aerosol is a polydisperse collection of well-mixed spheres. The spectra for 618 Pa nonane + 367 Pa D2O and 487
Pa nonane + 538 Pa D2O have been offset by factors of 100 and 10 respectively for clarity. (b) the reduced chi-squared plotted as a function of distance from the throat for the cases mentioned in (a). The worst fits correspond to droplets that form when the D2O concentration high and the nonane concentration is low...... 124
Figure 26 Fitting of the SAXS spectra to (a) a D2O core-nonane shell structure and (b) nonane core-D2O shell structure...... 126
Figure 27 Fitting of the scattering spectrum to a bimodal distribution of droplets (a)
Nonane and D2O volume fractions are fixed to those derived from FTIR while fitting (b) volume fraction of nonane is fixed and that of D2O is allowed to vary(c) volume fraction of D2O is fixed and that of nonane is allowed to vary...... 130
Figure 28 Fitting of the scattering data to scattering from (a) a sphere inside sphere structure (b) a sphere on sphere structure (c) a lens-on-sphere structure...... 132
Figure 29 D2O and reduced chi-squared for different contact angles of a lens-on-sphere model...... 133
xix Figure 30 Guinier plot for the test case of D2O (pv0=325 Pa) and nonane (pv0=625 Pa). The circles show the raw data and the line shows the fit through the linear section of the data points...... 134
Figure C 1 (a) and (b) The experimental and theoretical droplet temperatures for nonane at pv0 =489 Pa are calculated as described in the manuscript. The error bars on Td,exp are based on a 5% uncertainty in calculating the number density and a 5% uncertainty in calculating the condensation rate, dg/dt based on the subjectivity of the sigmoidal fit.
Td,exp is consistently higher than Td,exact and the absolute difference between them decreases as we move downstream. (c) The experimental and theoretical growth rates for nonane at pv0 =489 Pa. Even when Td is much higher than T the isothermal and non- isothermal growth laws predict essentially the same growth rate...... 284
Figure C 2 (a) and (b) The experimental and theoretical droplet temperatures for nonane at pv0 =322 Pa are calculated as described in the manuscript. The error bars on Td,exp are based on a 5% uncertainty in calculating the number density and a 5% uncertainty in calculating the condensation rate, dg/dt based on the subjectivity of the sigmoidal fit.
∆Texp is almost twice that of ∆Texact and this pv0 represents our worst disagreement. The
∆Texp at 2 µs is ~50 K and we think that it is a faulty reading and have not shown it here.
(c) The experimental and theoretical growth rates for nonane at pv0 =322 Pa. Even when
Td is much higher than T the isothermal and non-isothermal growth laws predict essentially the same growth rate...... 285
Figure C 3 (a) and (b) The experimental and theoretical droplet temperatures for D2O at pv0 =520 Pa are calculated as described in Section 3.3.2. The error bars on Td,exp are based on a 5% uncertainty in calculating the number density and a 5% uncertainty in calculating the condensation rate, dg/dt based on the subjectivity of the sigmoidal fit...... 286
xx Figure D 1 The experimental parameters for a case when nonane and D2O are condensing together (pv0= 357 Pa D2O +319 Pa nonane). a) The pressure and temperature along with the isentropic profiles. b) The average radius r and the spread σ in the droplet size distributions. c) The mass fraction condensed (g). The black symbols are g calculated from the FTIR vapor analysis. The white symbols are g calculated from the FTIR analysis from the liquid part of the spectrum. The dashed line is the incoming mass fraction of the vapor, gmax. The dark lines are gfit as mentioned in the integrated analysis section 2.3.4.d) D2O dominates nucleation in this case and the D2O nucleation pulse calculated from CNT is depicted by grey line and is normalized by its maximum value and is depicted on the left axis. The number densities calculated from SAXS are depicted along with the scaled nucleation values calculated from integrating the CNT nucleation ~ rates to match the maximum value of N SAXS . The data presented in this figure was first published in AIP conference Proceedings, Volume 157, pages-51-54 (2013)...... 288
Figure D 2 The experimental parameters when both nonane (pv0= 487 Pa) and D2O (pv0= 364 Pa) are condensing together. The symbols are explained in Figure D1. Since this is a case of competitive nucleation, the nucleation pulse is not shown here...... 289
Figure D 3 The experimental parameters when both nonane (pv0= 311 Pa) and D2O (pv0= 538 Pa) are condensing together. The symbols are explained in Figure D1. This is a case where D2O dominates nucleation...... 290
Figure D 4 The experimental parameters when both nonane (pv0= 622 Pa) and D2O (pv0= 547 Pa) are condensing together. The symbols are explained in Figure D1. This is a case where D2O dominates nucleation...... 291
Figure D 5 The experimental parameters when both nonane (pv0= 325 Pa) and D2O (pv0= 704 Pa) are condensing together. The symbols are explained in Figure D1. This is a case where D2O dominates nucleation...... 292
xxi Figure D 6 The experimental parameters when both nonane (pv0= 489 Pa) and D2O (pv0= 714 Pa) are condensing together. The symbols are explained in Figure D1. This is a case where D2O dominates nucleation...... 293
Figure D 7 The experimental parameters when both nonane (pv0= 621 Pa) and D2O (pv0= 718 Pa) are condensing together. The symbols are explained in Figure D1. This is a case where D2O dominates nucleation...... 294
Figure E 1 An illustration of the lens-on-sphere structure...... 298
xxii Chapter 1 Introduction
This chapter is an introduction to my general field of research. This chapter discusses the general background and the specific motivation for my research topic and an outline of the dissertation document.
1 1.1 Phase Transitions
Phase transitions are important processes in the environment and in industry. Some of the natural processes where phase transitions occur include evaporation of water from the seas and formation of clouds – a region in the atmosphere where water vapor, liquid and solid can co-exist. Phase transitions are also common in industrial processes, e.g. the semi-conductor industry uses chemical vapor deposition and atomic vapor deposition to fabricate features on silicon wafers. In the energy industry, coal-fired power plants produce steam from water which is then used to rotate turbines and consequently, generate electrical power. Phase transitions also serve to initiate separations. For example, a gas-vapor mixture can be chemically separated by condensing the vapor.
Studying the fundamentals of phase transitions is important for understanding these processes. Most of the research described in this thesis focuses on studying phase transitions from the vapor phase to the liquid phase, or condensation.
All first order phase transitions are three step processes. The first step, nucleation, requires a sufficient number of molecules to come together to form a critical nucleus – the first fragment of the new phase. The second step is growth where the remaining molecules of vapor condense on the nucleus causing an increase in its size. The final step is the ageing process where the clusters agglomerate to form larger structures and smaller particles evaporate while larger particles grow.
2 1.2 Classical Nucleation Theory
The critical step in vapor-liquid nucleation is the formation of a critical nucleus from the mother phase i.e. the reservoir of condensable vapor. The driving force for the nucleation process is the saturation ratio, S, defined as
p S v (1.1) pv,eq (T )
where pv is the partial pressure of the vapor and pv,eq(T) is the equilibrium vapor pressure at temperature T. S has to be greater than 1 for nucleation to take place. There is an opposing force to nucleation that arises from the work required to form the new surface separating the old and new phases. Classical nucleation theory (CNT) (Becker and
Doring, 1935) quantifies the balance between the forces driving and opposing nucleation in terms of the Gibbs free energy. CNT also assumes that the new cluster formed is spherical in shape, and the physical properties of the cluster – like surface tension and density – are the same as those of bulk liquid. This is called as the capillarity approximation.
The Gibbs free energy change ∆G during the process of nucleation can be written as
2 / 3 2 1/ 3 G nkBT ln S n (36vl ) , (1.2)
where n is the number of molecules in the cluster, vl is the molecular volume of the liquid, ζ is the vapor-liquid surface tension and kB is the Boltzmann constant. The first term on the right hand side in Equation (1.2) represents the decrease in ∆G due to the
3 formation of the new phase, and this term decreases linearly with n. The second term represents the increase in ∆G due to the formation of the new surface and this term increases as n2/3. The mathematical form of Equation (1.2) suggests that when S is greater than 1, there is a maxima in ∆G with respect to n. Figure 1 illustrates the shapes of these curves for different values at S assuming the physical properties of water at 298 K.
The value of n where this maxima occurs is the critical cluster size n* and the corresponding value of ∆G is called the Gibbs free energy change of the critical cluster,
∆G*. Table 1 mentions theses values of n* and ∆G* as a function of S for H2O at 298 K.
When a cluster reaches this size, addition of an extra molecule of vapor is favorable to the system, as ∆G begins to decrease. Equation (1.3) gives an expression for ∆G*.
1 G* n *2 / 3 (36v 2 )1/ 3 (1.3) 3 l
The principle of inducing condensation by increasing the saturation ratio can be used to remove condensible impurities from a gas stream. There are many processes which can increase S. One such process is adiabatically expanding the flow. The adiabatic expansion results in cooling of the mixture resulting S to increase rapidly. Once a high enough S is reached, the condensable vapor condenses to form droplets. A supersonic separator can provide for the adiabatic expansion of the gas vapor mixture in a continuous flow process capturing the condensate. We use supersonic nozzles as a replacement for industrial sized supersonic separators.
4 Table 1 It shows the values of n* and ∆G* for different values of S when water vapor is condensing at 298 K. We can see that as S increases, n* and ∆G* decrease.
S n* G *
kBT
1.1 185700 8856
5 39 31
10 13 15
Figure 1: ∆G as a function of n for different values of S calculated from Equation (1.2). We can see that as the value of
S increases, n* and ∆G* decrease demonstrating that the barrier to nucleation is lowered when S is high.
5 1.3 Motivation for the current work
Natural gas is an important source of energy that supplies around 30% of US energy needs (Department of Energy website). The main component of natural gas is methane and the composition of the impurities varies depending on the well location. The common associated gases include ethane, propane, butane and higher alkanes, and impurities like
CO2, H2S and water vapor. These impurities need to be removed before transportation so that we have pipeline quality natural gas. First, associated liquids which are mostly water and natural gas condensates are removed followed by acidic gases like H2S and CO2 which are removed by amine treatment. Water vapor is supposed to be removed after these processes. Conventional ways of water removal include glycol dehydration and pressure swing absorption which work by reducing the dew point temperature of the gas- vapor mixture by adding chemicals. These methods require large equipment and manned operation and there are environmental concerns regarding the disposal of associated waste. There is an alternate mechanical process to remove water vapor by using a supersonic separator (Schinkelshoek et al, 2006; Feygin et al, 2006).The separator condenses the vapor by cooling it. The condensed liquid consists of water vapor and higher alkanes and is removed using a cyclonic separator and rest of the gas is recompressed using a diffuser. Although this equipment causes a pressure drop of ~25%, its small size as well as the lack of moving parts and need for chemicals makes it a good choice for off-shore and subsea applications where unmanned operation is preferred. The supersonic separator combines two steps of natural gas processing-dehydration and dew
6 pointing into a single step which reduces the size of the plant. The supersonic nozzle used in this work does not have the cyclone separator or the diffuser section.
Previous work on alkane-water co-condensation has been done by the group of van
Dongen (Luitjen et al, 1998; Peeters et al, 2004). They measure the droplet sizes and their growth rates using constant angle Mie scattering. However, they were able to measure the droplet sizes only when the droplets are larger than 0.15 µm. The timescales of measurement are also of the order of milliseconds and not microseconds as found in the industrial supersonic separator. The homogenous nucleation rates from a mixture of supersaturated vapors of nonane and D2O were studied by Wagner and Strey (Wagner and Strey, 1990 and 2001) and Viisanen and Strey (Viisanen and Strey, 1996). These studies found that water and nonane nucleate independently and that the presence of one component in small amounts hardly affects the nucleation rate of the other component.
1.4 Objective and Thesis Outline
The objective of this work is to understand the fundamentals of droplet formation and growth of water, alkanes and their mixtures in supersonic nozzles. Droplet formation and growth in these conditions is interesting because hydrocarbons can wet water but not vice versa. In order to understand how the presence of higher alkanes affects the dehydration process, we need to study specifically how the nucleation rates and condensation of water are affected by the presence of higher alkanes. We also need to know how well the existing droplet growth law models predict growth rates and droplet temperatures when the time-scales are on the order of microseconds and the droplet sizes are on the order of
7 nanometers, conditions that are found in supersonic separators. We would also like to probe the microstructure of the composite aqueous-alkane droplets because the droplet’s surface composition could play an important role in determining their growth rates. The supersonic separator combines two steps of natural gas dehydration and natural gas dew pointing into a single step of removal of water vapor and higher alkanes together. It would be important to study how the presence of higher alkanes affects the dehydration process.
The organization of this thesis is as follows. Chapter 2 presents the experimental set-up and the different experimental techniques of Pressure Trace Measurements (PTM),
Fourier transform infrared Spectroscopy (FTIR) and Small Angle X-ray Scattering
(SAXS) used in this work.
Chapter 3 contains the experimental results and analysis related to the growth of pure component droplets. The experimental growth rates are studied using PTM and SAXS.
The experimental growth rates are compared to the predictions of a Hertz-Knudsen model that assumes either isothermal or non-isothermal droplet growth in the free molecular regime. For nonane, the predicted growth rates are insensitive to both droplet temperature and the evaporation coefficient, and agree well with the experimentally measured growth rates. For D2O, droplet growth rates are quite sensitive to droplet temperature, and the best agreement between experiments and theory are achieved for a condensation coefficient of 1 and an evaporation coefficient in the range from 0.5 to 1. Under our experimental conditions, incorporating coagulation is important to match the measured
D2O growth rates but not those of nonane. 8 In chapter 4, we study the co-condensation of nonane and D2O in a supersonic flow, using SAXS to characterize the size and number density of the aerosol and FTIR to determine the composition. Under our experimental conditions, we observe D2O nucleating at a rate that is somewhat reduced by the presence of nonane. This behavior can be qualitatively explained by non-isothermal effects during nucleation of the critical cluster. We also observe that nonane readily condenses on the water clusters impeding the growth of the nanodroplets by D2O. The D2O condensation is also partly inhibited by the hotter droplets which cause the D2O evaporation rate to increase to a comparable value of the D2O condensation rate.
Chapter 5 is the most challenging part of the thesis where we investigate the droplet structure of nanodroplets containing both nonane and D2O. Scattering from standard shapes like well-mixed spheres and core-shell structures does not agree with the experimental data. The ‘lens-on-sphere’ models suggested by molecular dynamics simulations fit the scattering data well, but the amount of D2O condensed, based on the
SAXS fitting parameters, is about 30-40% lower than that measured by infrared absorption spectroscopy. Chapter 6 summarizes the dissertation research and discusses the future work required to advance the field.
References:
1. Department of Energy website http://www.eia.doe.gov/forecasts/aeo/executive_summary.cfm.
2. R. Becker and W. Döring. Kinetische Behandlung der Keimbildung in übersättigten Dämpfern. Annals of Physics. (Leipzig) 24, 719-752 (1935). 9 3. C.C.M. Luitjen, R.G.P. van Hooy, J.W.F. Janssen and M.E.H. van Dongen, Multicomponent nucleation and droplet growth in natural gas. Journal of Chemical Physics. 109, 3553-3558 (1998).
4. Feygin,V; Imayev, S.; Alfyorov, V., Bagirov, L.; Dmitriev, L; Lacey, J.; “Supersonic Gas Technologies” presented at the 23rd World Gas Congress, Amsterdam, 5-9 June (2006). Available at: http://www.igu.org/html/wgc2006/pdf/paper/add11530.pdf.
5. P. Peters, G. Pieterse and M. E. H. van Dongen. Multi-component droplet growth. I. Experiments with supersaturated vapor and water vapor in methane. Physics of Fluids 16, 2567-2574 (2004).
6. P. Peters, G. Pieterse and M. E. H. van Dongen. Multi-component droplet growth. II. A theoretical model. Physics of Fluids 16, 2575-2586 (2004).
7. Schinkelshoek, P; Epson,H.; “Supersonic gas conditioning for NGL recovery”, presented at the 2006 Offshore Technology Conference, Houston Texas, 1-4 May (2006). Available at: http://www.twisterbv.com/download/paper_twister_OTC_may2006.pdf.
8. P.E. Wagner and R. Strey, Observation of Steady-State Aerosol Formation by Nucleation in Vapor Mixtures during Adjustable Expansion Pulses , Aerosols in Science, Industry and Environment Proceedings of the 3rd International Aerosol Conference, Kyoto, Edited by S. Masuda et al., 201-204 (1990).
9. P.E. Wagner and R. Strey. Two-Pathway Homogeneous Nucleation in Supersaturated Water-n-Nonane Vapor Mixtures. Journal of Physical Chemistry B 105, 11656-11661 (2001).
10 10. Y. Viisanen and R. Strey. Composition of critical clusters in ternary nucleation of water-n-nonane-n-butanol. Journal of Chemical Physics 105, 8293-8299 (1996).
11 Chapter 2 Experimental Methods
Parts of this chapter have been adapted from a manuscript recently accepted in The
Journal of Aerosol Science and Technology titled “Non-isothermal droplet growth in the free molecular regime” authored by Harshad Pathak, Kelley Mullick, Shinobu Tanimura and Barbara E. Wyslouzil. Section 2.3.2 is incorporated in a manuscript titled “Co- condensation of Nonane and D2O in a supersonic nozzle” which will be submitted for publication in The Journal of Chemical Physics. The co-authors are Barbara Wyslouzil,
Judith Wölk and Reinhard Strey.
12 2.1 Materials
In the industrial separation process, high-pressure methane acts as the carrier gas. High pressure methane is, however, expensive and raises significant safety concerns in an academic laboratory environment. Hence, a safer alternative nitrogen is used as the carrier gas in our experiments. We use heavy water (D2O) rather than water (H2O) because spectroscopy can accurately analyze D2O vapor concentrations within the nozzle without interference from vapor present in the ambient environment. Furthermore, if neutron scattering experiments are conducted to better constrain the droplet structure, then D2O is preferred because this compound has a much higher scattering length density than H2O. The representative higher alkane chosen is nonane for two reasons. First so that we can compare our results to those in the literature (Wagner and Strey,1990 and
2001; Viisanen and Strey,1996; Peeters et al, 2001) and second because methane/nonane mixtures have a similar vapor/liquid phase equilibrium curves as a mixture of natural gas
(Muitjens 1996).
We use liquid N2 from Praxair or Airgas that has a minimum purity of 99.99%, D2O
(Cambridge Isotopes Labs) that has more than 99.9% D substitution, and n-nonane
(Sigma Aldrich and Chemsampco) that is more than 99% pure. The thermophysical properties for these species used in the analyses are reported in Appendix B.
2.2 Experimental set-up
As illustrated in Figure 2, a carrier gas-vapor mixture is passed through a supersonic nozzle with the help of two rotary vane vacuum pumps which have a pumping capacity
13 of 0.13 m3/s. The nitrogen gas is drawn from the gas side of two liquid nitrogen Dewars, heated to room temperatures using in-line electric heaters, and the N2(g) flow is controlled by using two mass flow controllers. Heavy water and nonane, which are liquids at room temperature, are vaporized using homemade vaporizers and then mixed with a preheated carrier gas stream which helps disperse the liquid, and mix and dilute the vapor. The flow rate of the condensibles is controlled by peristaltic pumps and measured with weight balances. We check for complete vaporization by looking through a glass window located on top of the vaporizer. The gas-vapor mixture is then passed through the plenum where the final temperature adjustment is made with the help of a water bath. The supersonic nozzle is located downstream of the plenum. The cross-sectional area of the nozzle consists of three stages – a straight section, a converging section and a diverging section.
We measure the static pressure in the straight section of the nozzle where the velocity is
~10% of that of speed of sound. We derive the stagnation pressure from the static pressure by correcting for the speed and density of the gas. As the gas passes through the converging section, the velocity increases from subsonic to sonic, while the pressure, density and temperature begin to drop. As the gas flows through the divergent portion of the nozzle, the velocity becomes supersonic, the gas continues to expand and cool, and the vapor spontaneously condenses. Finally the aerosol is discharged to the atmosphere by the vacuum pumps.
The plenum and the nozzle are located on a linear translation stage which monitors the movement to an accuracy of 0.02 mm. The supersonic nozzles are Laval nozzles and are machined from aluminium with shaped top and bottom blocks and flat sidewalls. We can
14 assume that the flow in the divergent section is one-dimensional because the diverging angle for the nozzles is less than 30. The sidewalls have openings along the center line whose shape and window material is determined by the type of experiments to be performed. For small angle X-ray scattering (SAXS), the sidewalls have 1 mm wide slit and the windows covering the slit are 25 μm thick mica. For the PTM and the spectroscopic studies, the nozzles have 2 mm thick CaF2 windows on a slit that is 6mm wide. We call the nozzle used for PTM and spectroscopic studies C3 and the nozzle used for SAXS C2. Although the expansion rates of Nozzles C3 and C2 are the same - d(A/A*)/dx = 0.076 cm-1, they have slightly different throat areas. We therefore adjust the flow rates to maintain both the desired stagnation pressure and the partial pressure of the condensible at the nozzle inlet for the corresponding PTM and SAXS experiments.
15 Figure 2 Experimental set-up to generate aerosols of D2O, nonane and their mixtures in a supersonic nozzle.
2.3 Experimental techniques
2.3.1 Static Pressure Measurements (PTM)
Static pressure profiles are measured inside the nozzle with the help of a movable static pressure probe whose tip is in the subsonic region to avoid shocks. A platinum resistance thermometer (RTD) is used to measure the stagnation temperature inside the plenum.
After measuring the static pressure p, we use the 1-dimensional equations describing
16 mass balance, energy balance, momentum balance and the ideal gas law, together called the diabatic flow equations, to determine the remaining variables of temperature T, density ρ, velocity u, mass fraction of condensed material g, and area ratio A/A* where
A* is the minimum area inside the nozzle. The conundrum of having 4 equations in 5 unknown variables is solved by measuring a dry trace, i.e. an experiment using only the carrier gas that corresponds to g equal to zero. The dry trace yields the effective area ratio, (A/A*)dry. Next, we measure the pressure profile when condensable vapor is expanding along with carrier gas. We solve the diabatic flow equations assuming that
A/A* for this case is same as (A/A*)dry. This assumption is not entirely appropriate because heat released during the condensation compresses the laminar boundary layer
(Tanimura et al., 2005). When two components are condensing, there is an extra unknown variable which is the composition of the condensed liquid. In such a case, we initially assume that the liquid composition is same as the vapor composition. Even if these assumptions are not entirely accurate, the initial analysis gives us a preliminary estimate of T, ρ and velocity that we then revise using additional results from Fourier
Transform Infrared Spectroscopy (FTIR) and/or Small Angle X-Ray Scattering (SAXS) experiments in an integrated data analysis technique described later in this chapter. The
FORTRAN code used to calculate these parameters using p and (A/A*)dry as input is given in Appendix A1.
2.3.2 Fourier Transform Infrared Spectroscopy (FTIR)
PTMs determine the heat released to the flow. When one component is condensing, it is easy to relate the heat released to the amount of condensate using the latent heat of
17 vaporization. However, when more than one component is condensing, PTM do not yield the composition of the condensed liquid. An alternate techniques, such as FTIR is required to determine the vapor and liquid composition.
In this work, we use a Perkin Elmer Spectrum100 instrument to measure the absorption spectrum as a function of position in the nozzle by moving the nozzle (See Figure 3). The nozzle assembly is mounted on a movable plate with a minimum resolution of 0.02 mm, and typically measurements are made in 3 mm steps. The light emitted by a quartz halogen bulb inside the instrument emits ultraviolet, visible and infrared radiation which is passed through the nozzle with the help of 6 plane mirrors and 2 focusing mirrors. The focusing mirrors have a focal length of 20 cm and the beam width is about 4 mm. The intensity of the light that is returned to the FTIR instrument is detected by a liquid nitrogen cooled Mercury Cadmium Telluride (MCT) detector .
A background or empty nozzle spectrum Ie is measured by flowing only nitrogen through the nozzle and the sample spectrum Is is measured when condensible – carrier gas mixture is flowing. The ratio of the sample spectrum to background spectrum gives the transmission. We know that our largest droplets are about two orders of magnitude less than the wavelength of the infrared light. Thus, losses due to scattering are negligible
(Signourell and Reid, 2011). Other losses during transmission are cancelled in the ratio of sample to background spectrum. Consequently, the light extinction is due to absorption by the vapor/aerosol mixture inside the nozzle. We can write the measured absorbance, A as
18 I s A log10 (2.1) I e
-1 For n-nonane, we measure the absorbance between 2800 and 3000 cm , a region that corresponds to the C-H stretch. The resolution of the measured spectrum is 4 cm-1 and 32 sample and 32 background scans are taken. Although the vapor and liquid absorption bands overlap, we can assume that the measured absorbance is a linear combination of the spectra corresponding to vapor and the liquid (Modak et al. 2013). We establish the
molar absorptivities of vapor nonane,v by measuring several spectra upstream of condensation where the concentration of vapor is known. The liquid nonane spectra is determined by measuring a number of spectra at the nozzle exit where the concentration of vapor and liquid is known from the integrated analysis of SAXS+PTM as described later in this chapter. An equation for this process is
A cnonane,v nonane,v l nonane,l (2.2) cnonane,l l
A detailed description of this approach is given by Modak and co-workers (Modak et al
2013). After establishing values for molar absorptivities of the vapor nonane,v and the
liquid nonane,l (Modak et al), we calculate the contribution of each phase using the Beer-
Lambert law and linear regression
Anonane nonane,v cnonane,v l nonane,l cnonane,l l (2.3)
19 Figure 3 A schematic of the arrangement of FTIR Spectrometer to measure the absorption spectrum of mixture inside the nozzle.
where cnonane,v and cnonane,l are the concentrations vapor and liquid nonane, respectively, and l is the pathlength, i.e. the width of the flow. Mass balance is not enforced and, thus, for nonane, we get two estimates of the concentration of nonane – one from cnonane,v and one from cnonane,l.
This approach does not work for D2O because both the peak intensity and the peak location of the O-D stretch in the hydrogen bonded liquid between 2300 to 2700 cm-1 is sensitive to temperature (Bhabhe et al, 2013; Buch et al, 2004; Devlin et al, 2000; Devlin and Buch 2003; Schaff and Roberts, 1994 and1998). We instead focus on the O-D stretch of the vapor in the region that is not affected by the O-D stretch of the liquid phase, i.e.
20 -1 -1 between 2770 cm and 2890 cm . To determine the vapor phase D2O concentration, we use the Beer-Lambert law
AD2O,v D2O,v cD2O,v l (2.4)
where εD2O,v is the molar absorptivity of the D2O vapor and cD2O,v is the concentration of the D2O vapor.
As shown in Figure 4a, the value of D2O,v is determined by measuring several spectra upstream of condensation where the concentration of vapor is known accurately from the
PTM and the composition of the gas mixture, and then averaging. The amount of D2O condensed is determined from the amount of vapor entering the nozzle, the amount of vapor remaining in the gas phase and mass balance. Again, 32 sample and 32 background scans are measured at a resolution of 4 cm-1. This resolution is too coarse to observe the true line widths of the spectrum, but increasing the resolution further leads to fringes in the measured absorbance spectrum, due to the CaF2 windows. Fringes decrease the signal to noise ratio, which already is quite low because of low vapor concentrations.
Comparisons of D2O vapor concentration between this approach and the high accuracy tunable diode laser absorption measurements of Paolo Paci (Paci et al., 2004) showed good agreement as shown in Figure 4b. Figure 4b illustrates the agreement of D2O vapor mole fraction between FTIR and TDLAS for two conditions as D2O condenses in the nozzle. These two experiments were performed at a stagnation pressure p0 of 60 kPa and a stagnation temperature of 250C. Thus, the coarse resolution shown in figure 4(a) does not harm the reproducibility of the spectrum and the calculated the vapor concentrations
21 shown in figure 4(b). The spectrum measured for the co-condensing vapors of n-nonane and D2O are analyzed by combining these two approaches of pure nonane and pure D2O.
Figure 4 (a)The molar absorptivity D2O,v of D2O vapor calculated by measuring several spectra prior to condensation.
The temperature and the partial pressure of D2O are given in the legend. (b) The comparison of mole fraction of D2O vapor between the FTIR and TDLAS for two different pv0 of D2O.
2.3.3 Small Angle X-ray Scattering (SAXS)
To understand the distribution of condensate within the aerosol, we perform small angle scattering experiments to characterize the droplet size distribution and the volume fraction of the aerosol. In small angle scattering experiments the wavelength of light should be smaller than the particle size. Since our particle diameters are under 50 nm we require a wavelength of the order of a few nm. The volume fraction of our aerosol is also only of the order of 10-6 and such low volume fractions require high light intensity. To satisfy both conditions we use the high intensity X-ray sources available at a synchrotron.
22 We perform these experiments at the synchrotron at the Advanced Photon Source (APS),
Argonne National Lab (ANL). We use beamline BESSRC-CAT 12-ID-C. The 0.2 mm ×
0.2 mm beam of 12 keV (wavelength λ = 0.103 nm) X-rays has a wavelength spread,
Δλ/λ of 0.01%. The intensity of the scattered X-rays is measured by a 2-dimensional charged-coupled-device (CCD) detector. The sample-to-detector distance is 0.85 m for pure D2O droplets and 2.13 m for other cases. The nozzle and plenum are mounted on a sliding plate to make the axially resolved SAXS measurements. The relative position of the nozzle is known to better than 0.02 mm. The data reduction program provided by the
APS integrates the 2-D data to produce the 1-D spectrum of intensity (I) vs. the scattering vector (q) after correcting for spatial inhomogeneties. The scattering vector is defined as
4 q sin (2.5) 2
where θ is the scattering angle. At each nozzle position, scattering is measured as N2 alone flows through the nozzle (background) and as the N2 – condensible mixture flow through the nozzle (sample). The background spectrum is subtracted from the sample spectrum to yield the scattering spectrum from the aerosol. We assume that the aerosol is comprised of a polydisperse distribution of spheres that follow a Schulz size distribution
(Kotlarchyk and Chen, 1983). This is a good assumption for pure component droplets but not for mixed droplets of nonane and D2O as will be discussed in chapter 5.
The scattering from a Schulz size distribution of polydisperse sphere is well-defined and the APS data analysis program is used to fit the experimental spectra to extract the
23 average radius r , the spread σ, and the intensity as q goes to 0, I0. The intensity is converted to an absolute scale by matching the D2O volume fractions observed during the
SAXS experiments to those obtained from Tunable Diode Laser Absorption
Spectroscopy (TDLAS) experiments conducted under the same conditions (Paci et al.
2004; Tanimura et al. 2005). The aerosol number density, N is calculated from these fitting parameters using the following formula
2 3 I (Z 1)5 1 N 0 (2.6) 4 (Z 6)(Z 5)(Z 4)(Z 3)(Z 2) 6 2 r (SLD )
2 Here Z=[ r /σ] -1 and ΔρSLD is the scattering length density difference between the droplets and the surrounding gas mixture. ΔρSLD depends on the density of the condensate
ρl and, therefore, depends on the temperature and radius of the droplets through the
Young- Laplace equation which accounts for the density change of liquid droplets due to the curvature of the droplets
2 1 K (2.7) l l,0 r
where ρl,0 is the liquid density without considering the curvature effect. Finally, we obtain the mass fraction of the condensate, g from
4 3 (Z 2)(Z 3) g N r l (2.8) 3 (Z 1)2
24 2.3.4 Integrated data analysis
PTM are our preliminary measurements. The pressure profile measured for a condensing flow mixture is used along with (A/A*)dry to make the initial estimates of T, ρ and other flow variables. This approach assumes that the boundary layers that grow along the nozzle walls are stable against condensation – an assumption that is reasonable up to the region of rapid droplet growth. Beyond this point, the pressure increase associated with heat release can change the boundary layers and the mass fraction of condensate can be significantly underestimated near the nozzle exit (Tanimura et al., 2005).
The next step in the integrated analysis is to use the initial estimates of T and ρ to determine new values of g from SAXS or FTIR. When only one component is condensing, we use SAXS to supplement our PTM and refine the estimates of flow variables. When two components are condensing, we use FTIR to supplement our PTM and refine the estimates of flow variables. The values of temperature enter the SAXS analysis via the liquid density and, therefore, the scattering length densities of the droplets. The mass fractions obtained from SAXS, gSAXS, are combined with the mass fractions obtained from the PTM gPTM during the initial stages of condensation and a sigmoidal curve is fit to the combined data set to yield gfit. The flow equations are solved a second time, now using gfit and the condensing flow pressure as input data. This approach provides improved estimates of A/A*, u, T and ρ. The new values of T and ρ are used to reanalyze the SAXS data and further improve the estimates of g. The process is repeated until the temperature converges to within 0.5 K, and usually requires less than 3
25 iterations. The FORTRAN code used to calculate the experimental parameters using p and gfit from SAXS as input is given in Appendix A2.
In FTIR analysis, T and ρ enter the analysis from the concentration of the vapor and the liquid density. The condensate mass fractions for each species are derived from FTIR experiments, gFTIR and sigmoidal curves are fit through the data to yield gfit for each component. The values gfit for each species and the pressure measurements are used as input with p to solve the diabatic flow equations to yield better estimates of T, ρ, u and
A/A*. The new values are then used to refine the values of gFTIR, and the process is repeated until the solution converges. In the current experiments, analysis based on pressure measurements alone can underestimate the temperatures by up to 10 K and overestimate densities by ~4% compared to the more accurate values derived in the integrated analysis. The FORTRAN code used to calculate the experimental parameters using p and gfit of each species from FTIR as input is given in Appendix A3.
An important aspect of integrating the different techniques is to account for the difference between the location of the actual throat, the minimum in the flow area and the location of the physical throat, the minimum cross-sectional area inside the nozzle. The difference arises because of boundary layer growth along the nozzle sidewalls. In SAXS experiments, positions are measured with respect to the physical throat whereas in PTM positions are measured with respect to the actual throat. In the current experiments the actual throat is located 0.9 mm downstream of the physical throat and all measurements are referenced to the actual throat.
26 References:
1. Abraham, S. F. and Lester, H. High-Speed Machine Computation of Ideal Gas Thermodynamic Functions. I. Isotopic Water Molecules. Journal of Chemical Physics 22, 2051-2058 (1954).
2. A. Bhabhe; H. Pathak and B. Wyslouzil. Freezing of heavy Water (D2O) Nanodroplets. Journal of Physical Chemistry A 117, 5472-5482 (2013).
3. V. Buch; S. Bauerecker; J.P. Devlin; U. Buck; J.K. Kazimirski. Solid Water
Clusters in the Size Range of Tens-Thousands of H2O: A Combined Computational/Spectroscopic Outlook. International Reviews in Physical Chemistry 23, 375-433 (2004).
4. J.P. Devlin; C. Joyce; V. Buch. Infrared spectra and structures of large water clusters. Journal of Physical Chemistry A 104, 1974-1977 (2000).
5. J.P. Devlin; V. Buch. Ice Nanoparticles and Ice Adsorbate Interactions: FTIR Spectroscopy and Computer Simulations. In Water in Confining Geometries, Buch, V.; Devlin, J. P., Eds. Springer-Verlag: Berlin Heildelberg, 425-462 (2003).
6. M. Kotlarchyk and S. H. Chen. Analysis of small angle neutron scattering spectra from polydisperse interacting colloids. Journal of Chemical Physics 79, 2461- 2469 (1983).
7. V. Modak; H. Pathak; M. Thayer; S.J. Singer and B.E. Wyslouzil. Experimental evidence for surface freezing in supercooled n-alkane nanodroplets. Physical Chemistry Chemical Physics 15, 6783-6795 (2013).
27 8. M.J.E.H. Muitjens. Homogenous condensation in a vapour/gas mixture at high pressures in an expansion cloud chamber. Eindhoven University of Technology, PhD thesis (1996).
9. P. Paci; Y. Zvinevich; S. Tanimura and B.E. Wyslouzil. Spatially resolved gas phase composition measurements in supersonic flows using tunable diode laser absorption spectroscopy. Journal of Chemical Physics 121, 9964-9970 (2004).
10. P. Peters, G. Pieterse and M.E.H. van Dongen. Multi-component droplet growth. I. Experiments with supersaturated vapor and water vapor in methane. Physics of Fluids 16; 2567-2574 (2004).
11. P. Peters, G. Pieterse and M.E.H. van Dongen; Multi-component droplet growth. II. A theoretical model. Physics of Fluids 16, 2575-2586 (2004).
12. J.E. Schaff; J.T. Roberts. The Adsorption of Acetone on Thin Films of Amorphous and Crystalline Ice. Langmuir 14, 1478-1486 (1998).
13. J.E. Schaff; J.T. Roberts. Structure Sensitivity in the Surface-Chemistry of Ice - Acetone Adsorption on Amorphous and Crystalline Ice Films. Journal of Physical Chemistry 98, 6900-6902 (1994).
14. Fundamentals and Applications in Aerosol Spectroscopy, CRC Press, Taylor and Francis Group, eidted by R. Signorell and J. Reid (2011).
15. S. Tanimura; Y. Zvinevich; B.E. Wyslouzil; M. Zahniser; J. Shorter; D. Nelson; B. McManus. 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 122, (194304)1-11 (2005).
28 16. P.E. Wagner and R. Strey, "Observation of Steady-State Aerosol Formation by Nucleation in Vapor Mixtures during Adjustable Expansion Pulses", Aerosols in Science, Industry and Environment Proceedings of the 3rd International Aerosol Conference, Kyoto, Edited by S. Masuda et al., page 201-204 (1990).
17. P.E. Wagner and R. Strey. Two-Pathway Homogeneous Nucleation in Supersaturated Water-n-Nonane Vapor Mixtures. Journal of Physical Chemistry B 105, 11656-11661 (2001)
18. Y. Viisanen and R. Strey. Composition of critical clusters in ternary nucleation of water-n-nonane-n-butanol. Journal of Chemical Physics 105, 8293-8299 (1996).
19. B. Wyslouzil; J. Cheung, G. Wilemski and R. Strey. Small Angle Neutron Scattering from Nanodroplet Aerosols. Physical Review Letters 79, 431-434 (1997).
29 Chapter 3 The Growth of Nonane and D2O Nanodroplets
This chapter is a part of the manuscript titled “Non-isothermal droplet growth in the free molecular regime” authored by Harshad Pathak, Kelley Mullick, Shinobu Tanimura and
Barbara E. Wyslouzil which was first published in The Journal of Aerosol Science and
Technology, 47 (12), 1310-1324 (2013). The experiments mentioned in this chapter were performed by Harshad Pathak, Kelley Mullick and Barbara Wyslouzil. The analysis of the experimental data was performed by Harshad Pathak with guidance from Barbara
Wyslouzil.
30 3.1 Introduction
The spontaneous condensation of water and n-alkanes in high speed flows is relevant to a broad range of engineering applications from the expansion of steam in low pressure turbines (Moheban and Young 1985; Bohn et al. 2003; Gerber and Mousavi 2007; Dykas et al. 2007; Bakhtar et al. 2007; White et al. 1996) to the removal of condensible materials from raw natural gas (Muitjens et al. 1994; Looijmans 1995; Rijkers et al. 1992;
Luijten et al. 1998; Peeters et al. 2001; Okimoto and Betting 2001). The initial fragments of the new phase arise via homogeneous nucleation, and as these nanodroplets grow they rapidly deplete the vapor phase and add energy to the flow. Both heat addition and vapor depletion quench nucleation, and the coupling between these two processes establishes the initial size distribution of the aerosol. Since, for a fixed amount of condensible vapor, the aerosol number density determines the final average droplet size, droplet growth in the free-molecular regime is critical even when the final droplets are of micron size.
In our earlier work (Sinha et al. 2009) modeling the condensation of H2O/D2O in gently diverging supersonic nozzles, we developed a 1-D model to predict the aerosol size distribution near the nozzle exit. The model combined a single nucleation rate expression
– calibrated against available rate measurements – with 5 different growth laws including both isothermal and non-isothermal expressions. The results were then compared to the aerosol size distribution parameters determined by in situ small angle X-ray scattering experiments near the nozzle exit. Contrary to our expectations, the isothermal calculations were more successful at predicting the final droplet sizes and number densities. This was because non-isothermal droplet growth did not quench nucleation
31 rapidly enough, the aerosol number density was over predicted and, thus, the average droplet size was under predicted. Our observations were similar to those reported by
Young (Young 1982) who modeled pure steam condensation in supersonic nozzles. He found that he could only match both pressure traces and reported droplet sizes if the evaporation rate was lower than that expected under equilibrium conditions; i.e. the droplets were growing more quickly than expected.
The goal of the current work is to directly test growth laws in the free molecular regime against a more extensive experimental data set obtained in our supersonic nozzles. In particular, the current experiments follow the condensation of n-nonane or D2O from dilute vapor mixtures with position resolved static pressure trace (PTM) and small angle
X-ray scattering (SAXS) measurements. The unique aspects of our work include the position resolved particle size distributions and condensate mass fraction measurements, as well as the self-consistent analysis method that yields more accurate estimates for the flow properties than those based on pressure measurements alone. As discussed in the experimental section, we use these data to directly estimate both the average temperature of the droplets Td and the droplet growth rates. The experimental droplet temperatures
Td,exp are compared to the values derived from the implicit solution of the coupled mass and energy fluxes, Td,exact, as well as to explicit approximations based on the Gyarmathy
nd rd model and 2 and 3 order corrections to this model, Td,approx (Smolders 1992). The experimental growth rates are compared to those predicted using Td,exp and Td,exact and the
Hertz-Knudsen growth law. The role that coagulation plays in our measurements is also considered.
32 3.2 Droplet Growth Laws:
Droplet growth is a dynamic process. The net growth rate is governed by the difference between the rate at which vapor molecules are incorporated into the droplet and the rate with which monomers evaporate from the droplet. Since the condensing vapor releases heat, the temperature of a growing droplet is higher than that of the surrounding gas and the monomer evaporation rate increases above that for a droplet in thermal equilibrium with its surroundings.
The equations used to describe the coupled mass and heat transfer problem depend on the
Knudsen number, Kn = l / 2 r , where l is the mean free path of a vapor molecule and r is the average radius of the droplet. In the free molecular regime, r is significantly smaller than l and Kn is much larger than 1. In our experiments, Kn is always greater than 10, and thus, droplet growth is always in the free molecular regime.
The Hertz-Knudsen (HK) droplet growth model, dr v p p ( r ,T ) l q v q eq d (3.1) dt c e 2mvkB T Td is based on the kinetic theory of gases and describes droplet growth in the free molecular regime (Hill 1966). In Equation (3.1), t is time, mv is the mass of a monomer, kB is the
Boltzmann constant, T is the temperature of the vapor phase, vl is the molecular volume of the condensate, qc and qe are condensation and evaporation coefficients, respectively,
pv is the partial pressure of the vapor, and peq ( r ,Td ) is the equilibrium vapor pressure
33 above the drop of radius r at temperature Td. The Kelvin-Helmholtz equation relates
peq ( r ,Td ) to the physical properties of the condensate by,
2v p ( r ,T ) p (T )exp l p (T )exp Ke (3.2) eq d eq d eq d kBTd r
where peq (Td ) is the equilibrium vapor pressure over a flat surface at temperature Td, and
is the surface tension of the liquid. Finally, Ke = 2 vl/(kBTd r ) is the Kelvin number.
The condensation coefficient qc corresponds to the fraction of molecules impinging on the droplet that are incorporated into the droplet, while the evaporation coefficient is defined as the ratio of the actual evaporation rate to the theoretical evaporation rate. The values of qc and qe are most often taken as unity in order to simplify the analysis although there is no compelling experimental evidence to suggest that this is true for all conditions
– especially non-equilibrium conditions – and all substances (Young 1982; Marek and
Straub 2001).
For water, there is still significant uncertainty (Gajewski et al. 1974) regarding appropriate values for qc and qe despite extensive experimental and theoretical studies.
Experimental values of qc that range over three orders of magnitudes, from 0.001-1, have been reported since Rideal (Rideal 1925) conducted the first room temperature experiments in 1925 (Marek and Straub 2001; Davidovits et al. 2004; Mozurkewich
1986). More recent experiments, for temperatures between 238 K and 298 K, have reduced the range to 0.01-1 (Zagaynov et al. 2000; Beloded et al. 1989; Hagen et al.
34 1989; Shaw and Lamb 1999; Davidovits et al. 2006), and Marek et.al. suggest that condensation coefficients less than 0.01 indicate contamination of the water surface
(Marek and Straub 2001). Molecular dynamics studies, for temperatures between 300 K and 350 K, find the value of qc is close to unity (Morita et al. 2004; Vieceli and Tobias
2004; Tsuruta and Nagayama 2004), a result that is at odds with many experiments. The reported values of qe also range from 0.01 to 1 (Eames et al. 1997), and experimental challenges associated with determining qe again include surface contamination as well as accurate surface temperature measurements (Hickman 1965; 1966). Recent measurements on a train of 12 μm to 15 μm D2O droplets injected into vacuum (1.33 mPa), yielded evaporation coefficients of 0.57+0.06, for droplet surface temperatures 255
K - 295 K, where the latter were determined using Raman thermometry (Drisdell et al.
2008). Here a value of qe less than one was attributed to a kinetic barrier to evaporation that Transition State Theory suggests arises from librational and hindered translational motions at the liquid surface (Drisdell et al. 2008).
To our knowledge, there are no experimental measurements of qc and qe for nonane.
Molecular dynamics simulations of films (Xia and Landman 1994) found qc is ~0.9 for n- hexane and ~1 for n-hexadecane.
When vapor condenses on an existing droplet, the heat released due to condensation increases the temperature of the droplet relative to that of its surroundings. The heat dissipation rate from the droplet to the surroundings depends on the temperature difference Td-T, the latent heat of condensation L(T), and the capacity of gas-vapor mixture to absorb the heat. As the droplet temperature increases, so does the equilibrium 35 vapor pressure and, thus, the evaporation rate relative to a droplet in thermal equilibrium with its surroundings. The net result is that hotter droplets grow more slowly. Thus, growth laws can be broadly divided into two categories: isothermal growth laws – those that assume droplets are at the temperature of the surroundings – and non-isothermal growth laws – those where droplet temperature can differ from that of the surroundings.
For non-isothermal growth laws, calculating the droplet temperature can be critical to predicting the growth rates for droplets. As discussed in more detail in Section 3.3.1of this chapter, the equations describing heat and mass transfer to growing droplets are strongly coupled, and to determine the droplet temperature Td, these equations should be solved simultaneously. The process is simplified by introducing the wet bulb or quasi- steady state approximation, i.e. the assumption that there is an instantaneous balance in the heat and mass transfer processes between the droplet and its surroundings.
The wet-bulb approximation is valid here because the heat flux due to latent heat from the droplet is much higher than the internal heat flux within the droplet (Smolders, 1992).
The internal heat flux within the droplet is zero if there is no thermal gradient within the droplet. The characteristic time for thermal decay Δtint within a droplet is given by
[Carlslaw and Jaeger,1959,p.235]
r 2 tint 0.3 (3.3) al
where al is the thermal diffusivity of the liquid, and is on the order of picoseconds for our largest droplets. Since Δtint is small compared to the hundreds of nanoseconds required
36 for the environmental temperature to change by 0.1 K we can assume that the internal heat flux within the droplet is zero. But even with the quasi steady-state approximation, there are no exact analytical solutions for the non-isothermal growth equations and numerical methods are required. The computational cost of the numerical solution has motivated researchers to develop explicit approximations of droplet temperatures that relate Td to droplet size and the conditions prevailing in the surroundings. Analytical expressions are derived by linearizing the Clausius-Clapeyron equation (Mozurkewich
1986; Mason 1953; Wagner 1982; Barrett and Clement 1988; Vesala et al. 1990) or by assuming that there is little difference between the partial pressure of the vapor far away from the droplet and the equilibrium pressure over a drop (Gyarmathy 1963).
Finally, we note that the kinetic theory of gases is used to calculate the average molecular velocities when deriving Equation (3.1). In a non-equilibrium process, the bulk vapor velocity at the interface can affect these average molecular velocity distributions
(Scharge 1953; Mills and Seban 1967) and accounting for this effect modifies the growth rate expressions. Young (Young 1991), however, noted that the effect is easily nullified by small changes in the droplet temperature.
3.3 Droplet Growth Experiments
Droplet growth experiments were carried out using the equipment described in Chapter 2
The specific conditions are summarized in Table 2. Vapors of nonane or D2O were expanded with N2 in the supersonic nozzle. We performed the basic analysis of PTM along with the SAXS experiments which have been described in detail in Chapter 2.
37 PTMs were conducted in nozzle C3 whereas SAXS experiments were conducted in nozzle C2. The expansion rate of Nozzles C3 and C2 is d(A/A*)/dx = 0.076 cm-1. Our experiments are broadly divided into 2 categories based on the complexity of aerosol evolution. The easiest case is that observed for nonane where, as we will show later,
SAXS measurements confirm that droplet growth occurs in the absence of coagulation. In contrast, SAXS experiments indicate that during the D2O droplet growth experiments coagulation was always observed.
38 Table 2 A summary of the position resolved condensation experiments where p0, T0 and pv0 are the pressure, temperature and partial pressure of condensable vapor at the inlet to the nozzle. The carrier gas in all experiments was
N2.
Nonane D2O
p0 (kPa) T0 (K) pv0 (kPa) p0 (kPa) T0 (K) pv0 (kPa) 30.2 308 0.322 30.2 308 0.520 0.489 0.683 0.625
3.3.1 Nonane
Droplet growth experiments for nonane were conducted with three different inlet condensible partial pressures (pv0). For the highest pv0, Figure 5 summarizes (a) the pressure and temperature profiles together with (b) the droplet radii, (c) the mass fraction of condensate, and (d) the specific number densities and the normalized nucleation rates.
All results are presented as a function of time where time is related to position x through the velocity u and the relationship dx = udt. Here, t = 0 corresponds to the onset of condensation that we define as the point in the flow where the nucleation rate is maximized. In Fig 5(d) the normalized nucleation rates were calculated using Classical
Nucleation Theory using the expressions and physical property data presented by Ghosh et al. (2010).
39 30 (a) 260 Nonane p = 625 Pa T = 350C (b) 0.5 v0 0 25 240
0.4 20 220 0 fit (nm) (K)
p/p r T 15
0.3 200 , r
10 180 isentrope condensing 0.2 flow T 5 p/p0 160 1.2 (c) 0.10 1400 (d) 1.4e+18 g g g g S =p /p (T) SAXS max PTM fit v eq 1.0 1200 1.2e+18 0.08 0.8 1000 1.0e+18
J/Jmax ) 0.06 0.6 8.0e+17 -1 800 Nscaled CNT/ max (kg g S NSAXS/
J/J 6.0e+17 600 0.4
0.04 N/ 4.0e+17 400 0.2 0.02 2.0e+17 200 0.0 0.0 0.00 0
-100 -80 -60 -40 -20 0 20 40 60 -20 0 20 40 60 Time since onset (s) Time since onset (s)
Figure 5 Nonane droplet growth experiments for T0 = 35 C and pv0 = 625 Pa. (a) The measured pressures and estimated temperatures of the expanding supersonic flow. The temperature estimates are based on the integrated analysis described in Sec II.D that incorporates both the PTM and the SAXS data. (b) The average droplet radius and the spread of the droplet size distributions. The solid line in (b) is a 3 parameter sigmoid fit to the average radii (c) The mass fraction of the condensate g based on SAXS, gSAXS approaches the total mass fraction of condensible, gmax, near the nozzle exit. The supersaturation of the nonane vapor peaks near onset. (d) The normalized nucleation rate based on the classical nucleation theory illustrates that particle formation is localized in the nozzle. The predicted specific number ~ densities N are calculated by integrating the nucleation rate expression with respect to time, and then scaling these ~ values to match the experimental number densities at the nozzle exit. In the absence of coagulation N is conserved.
The specific number densities measured using SAXS follow the trend predicted by theory. The error bars in (c) and (d) represent the systematic uncertainty of ±5% that arises from the estimated uncertainty in the absolute calibration procedure for the SAXS experiments.
40 As illustrated in Fig. 5(a), the pressure and temperature decrease isentropically up to the onset of condensation. Beyond this point, the heat released by condensation increases the pressure and temperature above that expected for the isentropic expansion of a non- condensing gas mixture. In Fig. 5(b) ~ 9 nm radius droplets are first observed about 1μs after onset. The droplets grow rapidly, more than doubling in size in about 40 μs, before growth slows as the vapor is depleted. The spread of the droplet size distribution increases from ~4.5 to 5.0 nm for the first 20 μs and then remains almost constant. The aerosol polydispersity ( / r ), therefore, decreases from ~0.50 near onset to ~0.20 near the exit. Fig. 5 (c) illustrates the changes in mass fraction of condensate g (left axis) and the supersaturation S of nonane in the vapor phase (right axis). The slightly negative values of g prior to onset are thought to arise from nonane vapor phase heat capacities that are slightly lower than those predicted by available correlations (Ghosh et al. 2010).
Shortly after onset, the estimate for g based on PTM alone are significantly lower those based on SAXS. The deviation reflects the fact that the PTM approach cannot account for compression of the boundary layer due to heat released by condensation. The SAXS estimates are based on the SAXS analysis described in section 2.3.3, whereas the
PTM+SAXS estimates are based on the integrated data analysis described in section
2.3.4. The PTM+SAXS results are used to calculate the partial pressure of nonane remaining in the vapor phase, the gas mixture temperature and density, and the supersaturation. The latter increases as the expansion proceeds, reaching a maximum near onset and then decreasing rapidly as the temperature of the gas mixture increases. The
41 values determined by the integrated data analysis, together with the measured droplet sizes and the physical properties of the vapor and liquid, are then used to calculate the droplet temperatures and droplet growth rates. Finally, as illustrated by the symbols in ~ Fig. 5(d), during the first 30 μs, the experimental specific number densities N =N/ρ, increase rapidly from 5 1017 kg-1 to a maximum value of 1.4 1018 kg-1 and then remain ~ constant. The rapid increase in N indicates that new droplets continue to form even as existing droplets grow. This is consistent with the trends observed for the normalized nucleation rates, as well as the specific number densities calculated by integrating the nucleation rate expression and correcting these to match the experimental number ~ densities near the exit. The constant values of N for t >30 μs show that droplets do not coagulate during the ~40 μs available after nucleation stops and before the aerosol leaves the nozzle. A simple estimate of the expected coagulation rate can be made using (Hidy and Brock 1970; Pruppacher and Klett 1997)
~ dN K ~ coa N 2 (3.4) dt 2
where the coagulation coefficient Kcoa is given by
1/ 2 3k T 1/ 2 B Kcoa 4 r (3.5) l
(Seinfeld 1986; Seinfeld and Pandis 1998). The coalescence efficiency β is taken as 1 because the Van der Waals forces should guarantee that two droplets stick once they
42 collide (Okuyama et al. 1984). Solving Equation (3.4) predicts that in ~100 microseconds coagulation only decreases the specific number densities by about 3%.
In general, the rate of droplet growth is a strong function of the droplet temperature, Td.
One advantage of our data set is that we can directly estimate droplet temperatures based on experimental data alone, Td,exp (Tanimura et al. 2010) and compare these to the values calculated based on both an implicit theoretical expression (Td,exact) and explicit approximations (Td,approx) to Td,exact. For our small droplets, droplet temperatures adjust rapidly enough so that we can use the quasi-steady state or wet-bulb approximation to determine the droplet temperature by simultaneously solving the mass flux and heat flux equations. Thus,
= - Jq Jm L(Td ) (3.6)
where Jq and Jm are the heat and mass fluxes from a droplet to its surroundings, respectively, and L(Td) is the latent heat of condensation.
In the free molecular regime the heat flux, due to collision of the carrier gas N2 with a droplet, is given by (Kennard 1938).
1/ 2 8k T p k 2 B N2 B J q r c pN mN Td T , (3.7) m k T 2 2 2 N2 B where m is the mass of a N molecule, p is the partial pressure of N , and c is the N2 2 N2 2 pN2 specific isobaric heat capacity of N2 in the vapor phase.
43 To calculate Td,exp, we equate the heat transferred from the growing droplets to the measured latent heat released by condensation while accounting for the temperature change of the vapor (Tanimura et al. 2010). Thus,
~ dg dL(T) dT NJ L(T) fit g g c (3.8) q dt fit dt dt fit p(v)
where cp(v) is the specific isobaric heat capacity of the condensible vapor. Our integrated data analysis yields gfit(t), and L(T) is set equal to the latent heat of condensation of the bulk. Calculations show that droplet curvature effects change the latent heat of the smallest droplets by less than 2% (Tanimura et al., 2010) and hence, can be safely ignored. The major contribution to the heat dissipation rate is the first term, L(T)dg/dt, ~ and the other two terms account for less than ~5% of the total. Thus, dgfit/dt and N are the most important experimental parameters used to determine Td,exp via Equations (3.7) and (3.8).
To determine Td,exact, we calculate Jm using Equation (3.1)
m p p ( r ,T J 4 r 2 v q v q eq d (3.9) m 2k c e B T Td and combined equations (3.6), (3.7) and (3.9) to yield
1 k p c m B T T N2 pN2 N2 d,exact m m 2 N 2 v . (3.10) T L(Td,exact ) [qc pv qe peq ( r ,Td,exact )] Td,exact
44 This implicit expression for Td,exact is solved numerically.
In computational fluid dynamics codes, solving for Td,exact using Equation (3.10) can be too time consuming, and, thus, the implicit equation is simplified to yield the droplet temperature explicitly as a function of its size and the surrounding conditions (Mason
1953; Barrett and Clement 1988; Gyarmathy 1963). The most common assumption is that the transport coefficients and the latent heat of condensation are constant in the temperature range (T,Td) and can be evaluated at T. The Clausius-Clapeyron equation,
= 2 dpeq / dT Lmv peq / (RT ), is integrated using this assumption to obtain (Smolders 1992) the equation
peq (Td ) Lm T T ln v d 1 (3.11) peq (T) kBT Td T
Alternatively, Gyarmathy (1963) expressed Jm and Jq in terms of the driving forces and
the Nusselt numbers for mass, NuM , and heat, NuH , transfer, i.e.
q p ( r ,T ) q p D J 4 r 2 Nu e eq d c v mod (3.12) m M p 2 r N 2
k J 4 r 2 Nu (T T ) . (3.13) q H d 2 r where the Nusselt numbers are the ratios of the convective to diffusive transport rates, p= p p is the total pressure, and k is the thermal conductivity of the mixture. D is N 2 v mod the modified diffusivity of the binary mixture defined as
45 Dpmv Dmod (3.14) kBTm
where D is the binary diffusion coefficient of the vapor-gas mixture. Both Dmod and k are evaluated at an intermediate temperature Tm between the droplet and the surrounding gas mixture where (Hubbard et al. 1975).
1 T (2T T) (3.15) m 3 d
Gyarmathy then rewrote Equation (3.6) using Equations (3.12) and (3.13), assuming qc=qe=1, as
TkNu 1 p T peq ( r ,Td ) H S d 1 1 (3.16) Dmod LNu M S peq (T ) T pv
where the saturation ratio S is given by S=pv/peq(T). He also transformed the right hand side of this equation into a logarithm to yield
p ( r ,T ) p ( r ,T ) eq d eq d . (3.17) ln 1 pv pv
When peq ( r ,Td ) pv , ≈ 1 and Gyarmathy found the following explicit solution for
Td (Gyarmathy 1963)
1 T TkNu p p Lm d 1 (ln S Ke ) H v v (3.18) T Gyarmathy Dmod LNu M peq (T ) kBT
46 where Ke∞ is the Kelvin number evaluated at temperature of the surroundings. For notational convenience, Smolders((Smolders 1992) rewrote Equation (3.18) as
Td 1 1 f (S, Ke )C1 C2 (3.19) T Gyarmathy where
f (S, Ke¥ ) = ln S - Ke¥ , - = TkNuH p pv C1 , and (3.20) Dmod LNuM peq (T)
= Lmv C2 kBT
The exact Nusselt numbers can we obtained by equating Equations (3.9) and (3.12) for
NuM or (3.7) and (3.13) NuH. In the free molecular regime, the Nusselt numbers are directly proportional to droplet size and when p << p they can be approximated as v N2
r 2kBTm NuM D mv (3.21) c p r 2m pmix N2 mix 1 avg NuH k 2 mix kBTm
where c pmix and mix are the specific isobaric heat capacity and the heat capacity ratio of the vapor-gas mixture and mavg is the average molecular weight in the vapor-gas mixture.
47 p ( r ,T ) When p ( r ,T ) p , ln eq d in Equation (3.17) can be expanded using a eq d v pv
Taylor series around peq ( r ,Td ) = pv to get a more accurate estimate for . The value of
differs depending on where the Taylor series expansion is truncated. The values for the first three approximations are
1 1
1 Td 2 1 C1 1 (3.22) 2 T Gyarmathy 2 1 T 1 T 1 C d 1 C d 1 3 2 1 T 3 1 T Gyarmathy Gyarmathy
Here, the subscript on refers to the derivative where the Taylor series is truncated.
The final droplet temperature is then calculated using
Td,approx Td 1 1 /(1 ) (3.23) T T Gyarmathy where
Td C1( 1) C2 1 T Gyarmathy (3.24) C1 C2
Smolders (Smolders 1992) used the second order truncation to calculate the droplet temperatures and also set (1+ )-1 in Equation (3.23) equal to 1- . The latter is a good approximation for his experiments where the saturation ratio of water was always less
48 than 10. In our experiments, S is much larger than 10 and this approximation introduces unnecessary error. Thus, all of our calculations use Equation (3.23) and (3.24) and the values of given by in Equation (3.22).
The exact and approximate droplet temperatures are evaluated using the measured average droplet sizes and the conditions prevailing in the gas mixture based on the iterative data analysis.
240 Nonane pv =621 Pa 235 0.6 0 (K)
d 230 d
220 T =T T d exp T =T T 0.4 d d,exact 30 Gyarmathy Td=Td,exp Tapprox and 2 T and approx 3 0.3 Texact 20 (K) T - 0.2 d T growth rate (nm/ rate growth 10 0.1
(b) (c) 0 0.0 0 20 40 60 80 0 20 40 60 80 Time since onset (s) Time since onset (s)
Figure 6 Nonane droplet temperatures and growth rates for experiments conducted at T0 = 35 C and pv0 =625 Pa are calculated as described in the text. The legend for (a) and (b) are the same. The error bars on Td,exp are based on a 5% uncertainty in calculating the number density and a 5% uncertainty in calculating the condensation rate, dg/dt based on the subjectivity of the sigmoidal fit. (a) The experimental and theoretical implicit droplet temperatures. (b) The temperature difference between the droplets and the surrounding gas mixture. (c) The experimental and theoretical growth rates agree well even when Td is much higher than T. The isothermal and non-isothermal growth laws predict essentially the same growth rate.
49 For the nonane experiment conducted at T0 = 35 C and pv0 = 625 Pa nonane, Fig. 6(a) and
(b) illustrate the experimental droplet temperatures, Td,exp, the droplet temperatures calculated using the implicit method assuming qc=qe=1, as well as the difference between the droplet temperatures and that of the gas mixture, ΔT = Td-T for all of the methods considered. The hottest droplets are ~20-30 K warmer than their surroundings, and correspond to the smallest droplets that are growing rapidly during the initial stage of condensation. As droplet growth slows, ΔT decreases to ~2 K near the nozzle exit. The droplet temperatures calculated using different levels of approximation, Td,approx are ~2 K to ~10 K higher than Td,exact suggesting that the approximations are not accurate under the current experimental conditions. Indeed, the values of calculated using Equation
(3.22) are 1.6 to 3.3 times higher than the value ε =1 obtained from Equation (3.17) when
Td=Td,exact. We therefore conclude that when peq ( r ,Td ) differs significantly from pv , the best way to determine theoretical Td is by using the implicit method. Luo et al (Luo et al. 2006) reached a similar conclusion while modeling water droplet growth in an expansion wave tube at temperatures below 280 K and saturation ratios up to 10.
During the initial stages of condensation, i.e. within 10 μs of onset, Td,exp values are almost all ~5 K higher than Td,exact. During this time the polydispersity of the aerosol is high, i.e. greater than 33%. Since Td, exact is calculated using the average droplet size, we investigated whether polydispersity affects Td,exact, by calculating Td,exact for sizes around the average droplet size and determined volume weighted average Td,exact. Including polydispersity changed Td,exact by less than ±1 K from that calculated assuming a monodisperse aerosol. For nonane, this observation is reasonable because the equilibrium
50 vapor pressure over the droplets is always negligible compared to the partial pressure of nonane in the vapor phase. Thus, the vapor impingement rate greatly exceeds the evaporation rate from the droplet, and Equation (3.10) is essentially independent of droplet size.
Accurate estimates for the mass and heat fluxes are critical for calculating the droplet temperatures, and the ratio of the experimental and theoretical mass, or heat, fluxes is very close to the ratio of ΔTexp and ΔTexact. We estimate that there is an uncertainty of 5% ~ in determining N and about 5% uncertainty in determining dgfit/dt based on the subjectivity of the fit to the g(t) data. Together, this results in a 10% uncertainty in ΔTexp.
Experiments at the two lower nonane flow rates, Figs. C1 and C2 in Appendix C, yielded similar trends for ΔT, although the agreement between Td,exp and Td,exact is not quite as good. A more complete understanding of these discrepancies is the subject of future research.
Figure 6(c) illustrates the nonane droplet growth rates for qc=qe=1.To determine the experimental growth rates, we do not differentiate the experimental droplet size versus time data directly. Instead, we fit a sigmoidal curve to the r(t) data and calculated the
experimental droplet growth rates, d r fit /dt from the slope of the fit. We choose a sigmoidal function since, in the absence of coagulation, the radius should not increase indefinitely with time. The sigmoidal fits we chose included a Gompertz function, a logistic function, or any other type of a sigmoidal function with up to 5 parameters. The goal is to fit the data well over the observed range, not to extrapolate the data. Our
51 approach works well except during the initial stages of condensation where it is often difficult to calculate the slope, and hence the growth rate accurately because of the subjectivity involved in choosing the fit function. This uncertainty is reflected in the error bars illustrated in Figure 6(c).
To calculate the theoretical growth rates we use Equation (3.1) with qc=qe=1, and for the special case of isothermal growth we assume Td=T. Figure 6(c) shows that despite the wide range in droplet temperatures, all of the non-isothermal growth laws predict essentially the same growth rates and that these rates differ very little from the isothermal growth rate. The reason for this behavior is that even when the nonane droplets are 30 K hotter than the surroundings, the equilibrium vapor pressure over the droplets is still only
5% of the partial pressure of nonane in the vapor phase. Thus, the evaporation rate from the droplets is negligible, and the growth rates are largely determined by the impingement rate. Furthermore, all of the growth laws agree with the experimental data. Since aerosol polydispersity had only a limited effect on Td,exact, it does not affect the growth rates for nonane calculated using Td,exact. Similar results were obtained for the experiments conducted with the other two partial pressures of nonane, and the corresponding figures are available in Appendix C.
Because the evaporation rates do not affect the nonane droplet growth rates, we cannot constrain the evaporation coefficient for nonane. In contrast, any reduction in the condensation coefficient, qc from 1 would decrease the growth rate proportionally. Thus, our data for nonane are clearly consistent with qc = 1.
52 3.3.2 D2O
D2O growth rate experiments are more complex than those involving nonane because, as noted earlier, the D2O aerosols coagulate on the time scale of the experiments. Figure 7 summarizes experimental data for a typical case of D2O condensation.
For the dilute D2O-N2 mixture illustrated in Fig. 7(a), the pressures and temperatures initially follow the isentropic profile up to the onset of condensation before deviating abruptly as heat is added to the flow by the rapidly growing droplets. The first reliable
SAXS measurements are possible ~12 µs after onset, Fig. 7(b), and here the average droplet size is only 2.6 nm, i.e. significantly smaller than the first nonane droplets we observed. The droplet size increases rapidly to r 4 nm over the next 12 µs, and then more slowly over the next ~ 80 µs, reaching 5.3 nm at t = 105 µs. The spread of the size distribution steadily increases from 0.8 nm to 1.3 nm, while the polydispersity decreases from 0.3 to 0.24. The D2O droplets are significantly smaller than the nonane droplets formed under comparable conditions for two reasons. The first is that the molecular volume of D2O is only 11% of that of nonane, and the second is that the number densities of the water droplets is about 20 times higher than for nonane.
Fig. 7(c) illustrates that the rapid increase in condensate mass fraction mirrors the rapid temperature increase. The values of gPTM merge smoothly with the gSAXS data, but gPTM again underestimates the condensate mass fraction in the later stages of condensation. As growth slows, the values of g stabilize at a value of 0.014, about 14% below the mass
fraction of D2O vapor initially entering the nozzle, gmax 0.0162. The supersaturation S
53 increases rapidly reaching its maximum close to onset before decreasing again. Figure
7(d) summarizes the measured specific number densities along with the normalized nucleation rates where the theoretical specific number densities are scaled to match the maximum specific number density observed. When the SAXS measurements first detect particles, ~12 µs after the onset, nucleation is essentially complete. This is consistent with the normalized nucleation rate calculation that finds J/Jmax ≈0 for t >10 µs. Over the next
~100 µs, the specific aerosol number density decreases by 42% from the peak value of
3.2 1019 kg-1. Finally, for t > 60 µs the relatively constant values of g suggest that in this part of the expansion droplets are growing only by coagulation. Thus, droplet temperatures and condensational growth rates will only be calculated for t < 60 µs.
54 0.6 260 (a) D O p = 683 Pa T = 350C (b) 2 v0 0 5
0.5 240 4
fit 0.4 220
0
(nm) 3 (K) p/p T , r 0.3 200 2
condensing 0.2 isentrope flow 180 T 1 p/p0
120 (d) (c) g g g gfit SAXS max PTM S =pv/peq(T) 1.0
100 0.015 3e+19 0.8 80 ) -1 0.010 0.6 max 2e+19 g 60 S (kg J/J N/ 40 0.4 J/J 0.005 max 1e+19 Nscaled CNT/ 20 0.2 NSAXS /
0.000 0 0.0 0 -60 -40 -20 0 20 40 60 80 100 120 -20 0 20 40 60 80 100 120 Time since onset (s) Time since onset (s)
Figure 7 D2O droplet growth experiments for T0 = 35 C, pv0 = 683 Pa, and p0 = 30.2 kPa. (a) The measured pressure and estimated temperatures of the expanding supersonic flow. (b) The average droplet radii and the spread of the droplet size distribution. (c) The mass fraction of condensate g as determined by SAXS and PTM, and a fit to the combined data. The water vapor supersaturation peaks near onset. (d) The predicted normalized nucleation rates and specific number densities are compared to the measured specific number densities. The predicted number densities are scaled to match the maximum in the observed number densities. The decrease in number density is due to coagulation and coincides with a slow increase in particle size. The error bars in (c) and (d) represent the systematic uncertainty of ±5% that arises from uncertainty in the absolute calibration procedure for the SAXS experiments.
The difference in coagulation rates for nonane and D2O primarily reflects the difference in number densities in the two experiments. In fact, Kcoa calculated from Equation (3.5)
55 for the small D2O droplets is about 40% of Kcoa for the nonane droplets. The decrease in
Kcoa is, however, easily offset by D2O droplet number densities that are an order of magnitude higher than those for nonane, and the fact that coagulation rates are proportional to the square of the number densities and vary only linearly with Kcoa.
Finally, the decrease in specific number densities is about 3 times faster than that expected when Kcoa is estimated using Equation (3.5). The enhanced coagulation rate of water droplets is thought to arise from van der Waals interactions (Kerminen 1994;
Kerminen et al. 1991; Alam 1987) and the small size of the droplets (Marlow 1980;
Kennedy and Harris 1990).
Figure 8 summarizes the results of droplet temperature calculations and the condensational droplet growth rates for the D2O experiment illustrated in Fig.7, assuming qc=qe=1. As illustrated in Fig. 8(b), the ΔTexp values decrease from the maximum value of
34 K for the smallest droplets we can detect to ~0 K at ~106 µs. For the first ~60 µs after onset the values of Td,exp are significantly higher than the predicted droplet temperatures as illustrated in Figure 8(a). At longer times ∆Texp approaches zero consistent with droplet growth dominated by coagulation.
56 270 260 0.15 250 d
(K) 240 Eq (1) d 230 T Td=T 220
(a) s) 0.10 Td=Td,exact 210 Td=Td,exp Texp T 0.05 30 Gyarmathy Tapprox and 2
Tapprox and 3 0.00 (K) T 20 exact T - d growth rate (nm/ rate growth T -0.05 10
(b) -0.10 (c) 0 0 20 40 60 80 100 120 0 20 40 60 80 100 120 Time since onset (s) Time since onset (s)
Figure 8 The droplet temperatures and growth rates for D2O experiments at T0 = 35 C, pv0 = 683 Pa, and p0 = 30.2 kPa are calculated assuming qc=qe=1. The legend for (a) and (b) are the same. The error bars on Td,exp are based on a 5% uncertainty in calculating the number density and a 5% uncertainty in calculating the condensation rate, dg/dt based on the subjectivity of the sigmoidal fit. (a) The experimental and theoretical implicit droplet temperatures. (b) The temperature difference between the droplets and the surrounding gas mixture. (c) The experimental and theoretical growth rates. Using the values of Td,exp in Equation (3.1) yields negative growth rates suggesting the droplets are unstable.
Unlike the nonane experiments, the Gyarmathy model and the 2nd and 3rd order corrections to this model, all predict the same droplet temperatures, and, furthermore, these all agree with the implicit droplet temperature, Td,exact.
The corresponding growth rates are summarized in Fig. 8(c). The experimental rates decrease monotonically from 0.11 nm/µs for the smallest droplets to ~0 at 106 µs. The largest uncertainties again correspond to the smallest droplets where one source of uncertainty stems from the functional form chosen to fit the r versus t data. The
57 predictions of the isothermal and non-isothermal growth rates vary significantly, and for
0 μs < t < 30 μs, the experimental growth rates lie between the predictions of the isothermal growth law and the non-isothermal growth law using Td = Td,exact. At intermediate times, these two predictions lie significantly below the expected rate.
Finally, the differences between using Td,exact and Td,exp in Equation (3.1) are striking. In particular, when Td,exp is used, Equation (3.1) suggests that the droplets are not stable and should in fact evaporate.
The reason for the discrepancy between ΔTexp and ΔTexact, and consequently the differences in the predicted growth rates can be traced back to the fact that the heat flux per droplet measured in the experiments (Equation 3.8) is ~30% higher than that ~ predicted by theory. From the experimental viewpoint, uncertainty in N alone cannot ~ account for this difficulty. Increasing N by 30% would increase gSAXS above ginf near the nozzle exit, i.e. more D2O would condense in the nozzle exit than entered. Independent
FTIR (Fourier-Transform Infrared Spectroscopy) estimates of the vapor phase concentration of D2O suggest that g is underestimated at most by 5% near the nozzle exit, ~ i.e. N can only be increased by ~5%. If we examine the uncertainty in dg fit dt , reducing this quantity by 30% does not seem reasonable either, since in all cases the estimates of g from SAXS line up consistently with those of the PTMs. For D2O, polydispersity of the aerosol affects the calculated T by less than 2%. Given that the uncertainty in the experimental values is not large enough to account for the observed discrepancy we instead re-examine Equation (3.1). One way to shift the mass and energy balances is to
58 vary the values of qc and qe. Since we already assume qc = 1, the net mass flux cannot be increased by changing qc. The net mass flux can, however, be increased if we decrease the value of qe. As illustrated in Figures 9 (a) and (b) if we assume qe=0.5, the experimental and theoretical droplet temperatures and the corresponding growth rates are in much better agreement. This value of qe lies between the value qe = ~0.1 required by
Young (1982) to match steam condensation data in supersonic nozzles, also measured in the free molecular range, and the value qe = 1 expected in the equilibrium state.
40 0.15
d
Texp (K) 20 T Texact with qe=1 -
d 0.00 Texact with qe=0.5 T c d 10
growth rate (nm/ rate growth -0.05
(a) -0.10 0 (b)
0 20 40 60 80 100 120 0 20 40 60 80 100 120 Time since onset (s) Time since onset (s)
Figure 9 The droplet temperatures and growth rates for the same experimental conditions as Figures 3 and 4, but with qc=1 and qe=0.5 (a) Td,exact agrees better with Td,exp when qe=0.5. (b) The growth rates match the experimental values more closely when qe is reduced.
Despite the great improvement, experimental growth rates are still higher than those predicted by theory at intermediate times, i.e. 32 μs< t < 60 μs. This discrepancy arises because in the experiment condensation and coagulation are both important but Equation
59 (3.1) only describes condensation. To account for the effect of coagulation on droplet growth we follow the approach of Alonso et al. (Alonso et al. 1998) and treat coagulation and growth as independent processes, writing the overall growth rate as dr dr dr dt dt dt Cond Coag (3.25) where the first term is growth due to condensation and the second term is growth due to coagulation. To estimate the coagulation rate, we start with a volume balance
æ ö 1/3 N r = r ´ ç 1 ÷ (3.26) 2 1 è N2 ø where the subscripts 1 and 2 refer to the aerosol prior to and after coagulation, respectively, and where we will ignore the small effect introduced by changes in aerosol polydispersity.
Differentiating Equation (3.26) with respect to time and recalling that and are N1 r 1 constant yields
æ ö æ ö d r d r 1 1 dN ç 2 ÷= ç ÷ = - r 2 (3.27) è ø è ø 3 2 dt dt Coag N2 dt
= where N2 NSAXS is available from the experiments as a function of time and the rate of change of specific number densities ranges from -1.98 1023 kg-1s-1 at ~20 µs to -
1.2 1023 kg-1s-1 near the exit. Equation (3.25) can therefore be rewritten in terms of measurable quantities as 60 ~ d r d r 1 1 dN SAXS r ~ (3.28) dt dt cond 3 N SAXS dt
Figure 10 illustrates the droplet growth rates calculated using Equation (3.28) for qc=1 and qe=0.5 and Td= Td,exact. As illustrated in Fig. 10(a), the modified growth rate better predicts the experimental growth rates than when coagulation is ignored. Likewise, Fig
10(b) the D2O experiment for pv0= 520 Pa (see Table 2), incorporating coagulation leads to reasonably good agreement between the experimental data and the predicted growth rates. Nevertheless, to refine the range of qe values consistent with our data we forced the growth rates predicted by Equation (3.28) to match the experimental growth rates assuming Td = Td,exp. When 5 < T/K < 25, qe = 0.5 ± 0.1, and the values of qe increased as T decreased. For T < 5 K, qe approached 1.
61 Figure 10 (a) The non-isothermal growth rates predicted using Td,exact and that ignore or incorporate coagulation, are compared to the measured growth rates for the experiments with pv0=683 Pa D2O. Coagulation is only considered for t
> ~20 µs because coagulation rates cannot be accurately measured at earlier times. (b) The non-isothermal growth rates predicted using Td,exact and that ignore or incorporate coagulation are compared to the measured growth rates for pv0=520 Pa D2O.
3.4 Summary and Conclusions
Motivated by the earlier modeling work of Sinha et al (2009), we directly investigate the growth of water (2.9< r /nm <5.2) and nonane (6.2 < r /nm < 25.2) droplets in the free molecular regime. Droplets are produced in a supersonic flow and characterized using
SAXS. The SAXS results are combined with PTM in an integrated analysis scheme to determine the other properties of the flow including temperature, density, velocity and the
62 area of flow. The experimental droplet temperatures are estimated from an energy balance, whereas the experimental growth rates are measured by fitting the r(t) data and taking the slope. The theoretical growth rates are calculated using both isothermal and non-isothermal versions of the Hertz-Knudsen model. The droplet temperatures for the non-isothermal growth law are calculated three different ways: by an implicit calculation using Equation (3.10) that yields Td,exact, or by two approximate explicit expressions, Equation (3.23) and Equation (3.24), that yield Td,approx. In our initial analysis, we assumed the evaporation and condensation coefficient were both equal to 1.
For nonane droplet growth, the explicit approximations of Td vary widely and Td is best calculated using the implicit method. Generally, ΔTd,exp is 20-30% higher than the implicit calculations during the initial rapid stage of droplet growth, improving as droplet growth slowed. Under the current experimental conditions, the low equilibrium vapor pressure of nonane results in evaporation rates that are negligible compared to the condensation rates. Consequently, the growth rate is relatively insensitive to Td and qe.
For qc=qe=1, the isothermal and non-isothermal growth rates differ by less than 5% and both agree well with the experimental growth rates. Thus, our data are consistent with qc=1 but we cannot draw any conclusions regarding qe.
For D2O under our experimental conditions, growth rates are quite sensitive to Td and qe, while the values of Td,approx are within 2 K of Td,exact. During rapid droplet growth, the values of Td,exp are ~5-10 K higher than Td,exact and using Td,exp in the Hertz-Knudsen growth law suggests the droplets should be unstable. These difficulties can be reconciled
63 by decreasing the value of qe for D2O to 0.5± 0.1 when 5 < T/K < 25. Our values of qe lie between the value qe= 0.1 determined by Young in supersonic flow experiments with steam, and the value qe = 1 expected under equilibrium conditions. Incorporating coagulation into the growth calculation is important in the D2O experiments but not in the nonane experiments. Finally, as in the work of Sinha et al. (2009) D2O droplets grew more quickly than suggested by non-isothermal growth laws with qc = qe = 1. Our data, however, confirm that the rapidly growing nanodroplets are significantly hotter than the gas mixture. The latter conflicts with Sinha et al.’s 1-D modeling results for both isotopes of water, where isothermal growth laws yielded the best agreement between the predicted droplet size at the nozzle exit and the corresponding droplet size measurements. This conflict could potentially be resolved by undertaking a more sophisticated modeling effort. Models that incorporate 2-D effects, tracking subtle changes to the shape of the boundary layer induced by condensation, may be able to explain how nucleation is quenched more quickly than expected from heat addition by droplet growth alone.
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72 Chapter 4 Co-Condensation of Nonane and D2O in a Supersonic Nozzle
This chapter forms the basis of a new manuscript titled “Co-condensation of Nonane and
D2O in a supersonic nozzle” that is submitted for publication in The Journal of Chemical
Physics. The authors for this manuscript are Harshad Pathak, Judith Wölk, Reinhard
Strey and Barbara Wyslouzil. The experiments mentioned in this chapter were performed by Harshad Pathak, Barbara Wyslouzil and Judith Wölk. The data analysis was performed by Harshad Pathak with guidance from Barbara Wyslouzil.
73 4.1 Introduction
Natural gas is a widely utilized energy source that is found in underground porous rock formations, associated with other hydrocarbon reservoirs in coal beds, or as methane clathrates in the deep oceans (Colwell et al. 2008; Reed et al. 2002). The water vapor present in most raw natural gas can interact with methane to form hydrates at temperatures above the freezing point of pure water, and thereby plug transportation pipelines. Condensation of higher alkanes in a pipeline can also lead to two-phase flow and increased pressure drop. Thus, water and other condensible impurities must be removed before the gas can enter most pipeline systems.
A recent technological innovation in dehydration technology consists of cooling the raw natural gas by expanding it in a supersonic nozzle and then capturing the condensed water/hydrocarbon droplets that are formed in the flow. Co-condensation of water and hydrocarbons is complicated by the fact that these species are highly immiscible and the question arises how the miscibility gap in the macroscopic phase influences the microscopic nucleation behavior and subsequent droplet growth.
Strey and co-workers (P.E. Wagner and R. Strey 1990; Viisanen & Strey 1996; Wagner and Strey 2001) were the first to systematically investigate these issues by conducting homogeneous nucleation in supersaturated vapor mixtures of n-nonane and water in a nucleation pulse chamber. They reported that although the nucleation in the water / n- nonane vapor mixture is essentially the superposition of two simultaneous unary nucleation processes, there was a slight increase in the onset activity of water required to maintain a constant nucleation rate as the concentration of n-nonane increased. The 74 concentration of the growing droplets was determined using constant-angle Mie- scattering (CAMS) (Wagner 1985), and, based on the differences apparent in the Mie scattering curves of the pure components, they concluded that even when nucleation was initiated by water at low nonane concentrations, the scattering curves corresponded to those of pure nonane droplets. Thus, even when water was the species that nucleated, the droplets that grew were rich in nonane. Wagner and Strey (Wagner and Strey 2001) suggested that the observed increase in water activity upon the addition of nonane could be due to a more rapid increase in the system temperature, caused by the growth of the newly nucleated droplets by n-nonane, over any change in the system temperature when only water condensed. Since an increase in temperature decreases the supersaturation, thereby reducing the nucleation rate, their explanation is equivalent to the presence of n- nonane decreasing the characteristic time for nucleation.
In 2001, Peeters et al. (Peeters et al. 2001) published their nucleation results for the ternary system methane - n - nonane - water measured in a pulse expansion wave tube.
They found that the nucleation rates measured in the ternary system were close to the sum of the nucleation rates measured for the two binary systems (methane / n-nonane and methane / water), and concluded that the presence of the second vapor did not influence the nucleation of the first, and that co-nucleation did not occur. Multi-component droplet growth in this ternary system was later studied by Peeters et al.(Peeters et al 2004). Here the experiments showed that the presence of water vapor did not affect the growth rate of droplets formed when n-nonane controlled nucleation, while the presence of n-nonane greatly increased the growth rate of droplets when water controlled nucleation.
75 Nucleation in the water / n-nonane system was also investigated by Chen et al. (Chen et al. 2003) using Monte Carlo simulations. They found that the critical clusters contained only water or n-nonane for all combinations of gas phase activities, and they interpreted their results as a nucleation process that follows two separate nucleation pathways. Even for clusters containing fewer than 15 molecules, the concentration of pure clusters was orders of magnitude higher than that of the mixed clusters. Finally, their simulations agree very with the critical onset activities for water / n-nonane obtained from Wagner and Strey (Wagner and Strey 2001).
Although the published data and simulations provide a coherent picture of particle formation and growth in this highly non-ideal system, our current supersonic nozzle experiments directly complement and extend the previous work in a number of different ways. In particular, our experiments probe the formation and growth of much smaller nanodroplets produced when dilute carrier gas/vapor mixtures are cooled at rates of ~105-
106K/s, conditions comparable to those found in industrial supersonic separators (Prast et al 2006). Furthermore, we directly follow the evolution of the drop size distribution in the nanometer regime using small angle X-ray scattering (SAXS), and independently measure the composition of the vapor and liquid phase using Fourier Transformed
Infrared (FTIR) absorption spectroscopy. The latter provides deeper insight into aerosol evolution than size measurements based on SAXS alone.
Our results confirm that the presence of nonane vapor inhibits D2O nucleation and condensation even in experimental conditions where D2O dominates nucleation. Under our conditions, nucleation rates decreases by up to a factor of 6 relative to the nucleation 76 rates of pure D2O. The subsequent condensation of D2O is inhibited in at least two ways.
First, during binary condensation the growing droplets are hotter than during pure D2O condensation and thus the evaporation rate of D2O from the droplets increases considerably and approaches the D2O impingement rate. Second, the structure of these droplets – with a surface comprised largely hydrophobic nonane (Pathak et al, AIP conference proceedings 2013) – is also expected to reduce the ability of water to be accommodated into the growing droplet (Chen et al 2003) . As in the pure droplet condensation studies (Pathak et al, Aerosol science and Technology 2013), nonane evaporation rates are negligible compared to nonane impingement rates and, thus, nonane condenses readily. For the single case where nonane appears to dominate nucleation, the presence of D2O increases the nucleation relative to that of pure nonane. Although the onset of condensation is determined by the high saturation ratios reached by the nonane vapor, the nucleation rate of D2O under these conditions is still estimated to be within a factor of 2 of the nonane nucleation rates.
4.2 Experiments and their analysis
The experimental set-up and techniques are described in Chapter 2. We perform PTM,
FTIR and SAXS to collect the data presented in this chapter. We completed experiments for three stagnation partial pressures pv0 of n-nonane, and three stagnation partial pressures of D2O. In total we conducted six unary experiments and nine binary experiments which were combinations of unary experiments. The conditions are
77 summarized in Table 3. In all cases the carrier gas was N2 and experiments started from a
° stagnation pressure p0=30.2 kPa and a stagnation temperature T0 = 35 C. TJmax is the temperature of the mixture corresponding to the maximum nucleation rate, and gmax represents the mass fraction of the vapor before condensation.
78 Table 3 All experiments used nitrogen as the carrier gas. TJmax and gmax values are given in this table. The values of pv0 are nominal values and the actual values are within 20 Pa of these values and are mentioned in Appendix D.
n-nonane pv0 = 0 Pa pv0 = 320 Pa pv0 = 490 Pa pv0 = 620 Pa
D2O TJmax(K) gmax TJmax(K) gmax TJmax(K) gmax TJmax(K) gmax
pv0 = 0 Pa 196.4 0.047 205.0 0.070 209.6 0.088
pv0 = 360 Pa 205.2 0.0082 205.2 0.055 206.3 0.058 210.4 0.064
pv0 = 540 Pa 212.1 0.0124 212.0 0.078 212.0 0.082 212.7 0.087
pv0 = 700 Pa 216.9 0.0162 215.9 0.096 216.3 0.100 215.9 0.104
4.2.1 Unary Condensation and Nucleation
Figure 11 summarizes the data for the three experiments in which pure n-nonane vapor condenses from a dilute carrier gas-vapor mixture. In Figure 11(a) the pressure ratio is on the left axis and the temperature on the right axis. The temperature is calculated using the integrated analysis summarized in the experimental section. Time t is determined from the position x and the velocity inside the nozzle using dt = dx/u where t = 0 corresponds to the throat of the nozzle. The dashed lines are the calculated isentropic profiles, and initially, the pressure and temperature decrease isentropically as the gas flows downstream. For n-nonane, the isentropic profiles do not overlap and as the amount of vapor increases, the expansion rate decreases. This is because adding n-nonane vapor to the carrier gas decreases the heat capacity ratio of the mixture, from 1.4 for pure N2 to
1.34 for the highest nonane pv0. A lower value of results in a slower expansion for the same A/A*. At the onset of condensation, the measured pressure and temperature profiles start to deviate from their respective isentropes as latent heat is released to the flow by the vapor to liquid phase transition. At higher pv0, onset moves further upstream and TJmax are 79 196 K, 205 K and 210 K for pv0 of n-nonane equal to 322 Pa, 489 Pa and 625 respectively.
Figure 11(b) shows the position resolved values of average radii r and the spread σ of the size distribution measured using SAXS. White symbols correspond to data for the lowest pv0, grey symbols to the intermediate pv0, and black symbols to the highest pv0.
This color scheme is followed throughout Figure 11. The droplets grow as they move downstream of the nozzle, following a sigmoidal curve with respect to time, and increasing pv0 leads to bigger droplets. The droplet sizes at the nozzle exit are r = 25
nm for the highest pv0 and r = 13 nm for the lowest pv0. A detailed analysis of droplet growth for the pure components is presented in the previous chapter. In contrast, the spread σ for the droplet size distribution is almost constant after increasing for about 20
μs after the first measurement and is about 5 nm for the largest droplets, 4 nm for the intermediate droplets and 3 nm for the smallest droplets.
Figure 11(c) illustrates the behavior of the mass fraction of condensate. The symbols are the mass fractions calculated from the SAXS analysis, gSAXS where the error bars represent a 5% uncertainty in our calibration factor when conducting SAXS experiments.
The dashed line is the overall mass fraction of the condensible in the flow, gmax and is the upper limit for g. The value of gmax increases from 0.047 for the lowest pv0 to 0.088 for the highest pv0. The line that closely follows the symbols is the mass fraction calculated from the integrated analysis of the PTM and SAXS data, gfit as described in the
80 experimental section. For the lowest pv0, there is not enough time for the condensate to approach its maximum value unlike the experiments at higher pv0.
In Figure 11(d), the left axis corresponds to the normalized nucleation rates J/Jmax calculated using classical nucleation theory (CNT), where Jmax is the maximum value of
J, and the right axis indicates the number densities normalized by the density of the flow, ~ ~ N/ρ= N . N SAXS are the number density measured from SAXS experiments whereas
~ N scaledCNT are the number densities calculated by integrating the nucleation pulse over
~ time and scaling them to match the maximum value of N SAXS The maximum nucleation
~ rate is reached just downstream of the onset, and in the absence of coagulation N SAXS is constant after nucleation is complete. The nucleation rates suggest that particles are ~ formed in a sharp pulse that lasts ~20 μs. The values of N SAXS show that the number of particles produced increases rapidly for ~20 μs after the first SAXS measurement but then remain constant. The dashed lines are the number densities calculated from CNT ~ that are scaled to match N SAXS at the exit. The number densities decrease with increasing
18 -1 18 -1 pv0 and are 410 kg for the smallest pv0 and 1.410 kg for the highest pv0. The fact ~ that N SAXS reaches a stable value means that the n-nonane aerosols do not coagulate on the time scale of these experiments.
81 0 Nonane p0=30.2 kPa T0=35 C
Figure 11 The experimental measurements for pv0 of n-nonane equal to 322 Pa, 489 Pa and 625 Pa. (a) the pressure ratio and temperature for the mixture as it expands in the nozzle. The dashed line is the isentrope and the solid line is the measured pressure or temperature. (b) the average radii, r and the spread σ of the droplet size distribution. (c)
the mass fraction, g of the liquid as it condenses in the nozzle. The dashed line is the maximum value gmax, the symbols, gSAXS are calculated from SAXS analysis and the lines which closely follow the symbols are calculated from the integrated data analysis, gfit. (d) The normalized nucleation rates on the left axis and the number densities on the right axis. The dashed lines are the number densities predicted by classical nucleation theory but scaled to match the number ~ densities at the exit. The symbols are number densities from the analysis of SAXS data N SAXS .
82 Figure 12 summarizes the results for the unary D2O condensation experiments, in a manner analogous to Figure 11. Figure 12(a) shows the pressure ratios and temperature profiles, and, unlike n-nonane, here the isentropic temperatures or pressure ratios overlap almost perfectly because of the mixtures is always very close to 1.40 – the value for pure N2. TJmax varies from 205 K to 217 K – temperatures higher than those for similar n- nonane partial pressures. Figure 12(b) shows that the droplets grow rapidly at first and then the growth rate decreases. The D2O droplets are much smaller than the nonane droplets with the values near the nozzle exit of r = 5.2 nm for pv0=683 Pa and r =
3.8 nm for pv0=346 Pa. This is due to the fact that although we are condensing comparable molar quantities of condensate, the volume of a nonane molecule is almost nine times that of a D2O molecule. Figure 12(c) illustrates the mass fraction condensed where the symbol convention matches that used in Figure 12(c). The amount of condensate increases rapidly after the onset but reaches a constant value near the nozzle exit, slightly below the maximum value, gmax. This suggests that the slow droplet growth observed in Fig 11(b) is driven by coagulation. Figure 12(d) shows that the duration of the nucleation pulse for D2O is shorter than that for n-nonane, ~14 μs for the lowest pv0 and ~11 μs for the other two experimental conditions. Furthermore, the number densities ~ determined by SAXS, N SAXS follow a similar trend for all three values of pv0. At the end
~ 19 -1 19 -1 of the nucleation pulse, N SAXS = 3 - 3.2 10 kg , decreasing to ~ 2 10 kg near the nozzle exit due to coagulation. As can be seen from Chapter 3, coagulation occurs in the
D2O experiments but not in the n-nonane experiments because the D2O number densities are high enough and the coagulation rate is fast enough to be observed in the short 83 timescales in the nozzle. The dashed lines are the calculated number densities from the time-based integration of J from CNT and are scaled to match the maximum number densities reached in the nozzle.
84 0 D2O p0=30.2 kPa T0=35 C
Figure 12 Experimental results for pv0 of D2O equal to 346 Pa, 520 Pa and 683 Pa. The black symbols correspond to the highest pv0, white symbols correspond to the lowest pv0 and the grey symbols correspond to the intermediate pv0. (a) the pressure ratio on the left axis and the temperature on the right axis. The dashed line represents the isentrope calculation.
(b) average radii, r and the spread σ of the size distribution. (c) mass fraction, g of the condensate. The dashed line
is the maximum value gmax, the symbols, gSAXS are calculated from SAXS analysis and the lines which closely follow the symbols are calculated from the integrated data analysis, gfit. (d) scaled nucleation rates on the left axis and the ~ ~ measured ( N SAXS ) and calculated ( N CNT ) number densities on the right axis. The calculated number densities are scaled to match the maximum number densities.
85 To ensure that the current experiments are consistent with our previous results we quantified the nucleation rates for these experiments and compare them to experiments completed in similar set-ups (Ghosh et al. 2010; Kim et al. 2004). The experimental nucleation rates can be derived from SAXS experiments using the following equation:
NSAXS Jmax J max (S,T ) (4.1) t J max vv
where NSAXS is the number density, vv is the density of gas in the viewing volume and
Jmax is the density of gas at the maximum nucleation rate. The second term on the right hand side corrects for any change in the density of the gas between the nucleation zone
and the viewing volume. tJ max is the characteristic time for nucleation calculated from
CNT by (Kim et al. 2004)
Jdt t J max (4.2) J max
Equation (4.1) does not incorporate coagulation and hence NSAXS should be measured before coagulation has reduced the number density, i.e. as soon as the nucleation pulse is over. For pure n-nonane, coagulation is negligible but for pure D2O, coagulation reduces number densities by ~30% by the nozzle exit.
86 Figure 13 The comparison of measured nucleation rates with previous work for a) Nonane and b) D2O. Temperature at the maximum nucleation rate, TJmax is mentioned next to the symbols.
As illustrated in Figure 13, under the current experimental conditions the nucleation rates
16 -3 -1 17 -3 -1 for n-nonane are of the order of 10 cm s and those for D2O are 10 cm s . n-Nonane nucleation rates are highly consistent with the earlier measurements of Ghosh et al. where separate experiments were performed using N2 and Ar as the carrier gas. The nucleation rates for D2O are also consistent with the earlier work of Kim et al. even though Kim et al. used Small Angle neutron scattering (SANS) rather than SAXS, and measurements were all made near the nozzle exit. Coagulation upstream of the viewing volume may explain why the nucleation rates measured by Kim et al are slightly lower than those reported in this work. Considering the differences in the experiments of Kim et al. from those described in this work, it is remarkable that there is such good agreement in the nucleation rates.
87 4.2.2 Binary Condensation and Nucleation
The analysis of binary co-condensation experiments is intrinsically more challenging than the analysis of the unary experiments both because the composition of the condensate does not equal the composition of the vapor phase (Tanimura et al. 2010)and because the droplet structure may not be spherically symmetric (Li and Wilemski 2006). We first address our efforts to measure aerosol composition and then briefly discuss the issues associated with droplet structure.
1.0 1.0 PTM integrated analysis integrated analysis 0.8 FTIR vapor 0.8 FTIR FTIR liquid PTM 0.6 0.6 max max g g / /
g 0.4 0.4 g
0.2 0.2
(a) 0.0 0.0 (b) 0 50 100 150 200 0 20 40 60 80 100 120 140 160 time (s) time (s)
Figure 14 The comparison of FTIR and the integrated analysis and PTM when (a) pv0= 625 Pa of nonane is condensing and (b) pv0= 683 Pa of D2O is condensing.
To follow the liquid and vapor compositions during binary condensation we turn to vibrational spectroscopy. Our earlier work with pure nonane (Modak et al. 2013) proved
88 that the mass fraction of condensate measured by FTIR spectroscopy is very close to that determined from the integrated analysis from SAXS+PTM , and both agree with the values estimated from PTM during the early stages of condensation which is illustrated in
Figure 14(a). Figure 14(b) shows that the same is true for D2O. As illustrated in Figure
15, during binary condensation the spectra exhibit features corresponding to the vapor and liquid phases of both nonane and D2O. Given the limited miscibility of these species, the presence of one compound should not affect the molar absorptivity of the other even in the condensed phase. To simplify the analysis we also avoid analyzing regions of spectral overlap. For nonane, the C-H stretch region between 2800 and 3000 cm-1 is not affected by the O-D stretch of D2O and can be analyzed using the methods developed for unary nonane condensation. For D2O, the O-D stretch of the vapor in the region between
-1 2770 to 2795 cm is not influenced by the O-D stretch of liquid D2O or the C-H stretch of nonane. Thus, we can fit the measured spectra in that region to determine the D2O vapor concentration, and then use mass balance to derive the condensate mass fraction of
D2O. We do not use the signal from the O-D stretch of the condensed phase because the intensity of this signal depends both on concentration and temperature (Bhabhe et al.
2013; Devlin and Buch 2003; Devlin et al. 2000; Buch et al. 2004; Schaff and Roberts
1998; 1994).
89 -3 30x10 D2O pv0 = 704 Pa Nonane vap+liq 25 Nonane pv0= 319 Pa x= 34 mm T=229 K 20
15
A D2O 10
5 liq vap
0 2400 2600 2800 3000 -1 wavenumber (cm )
Figure 15 A typical IR spectrum during co-condensation of D2O and nonane. The vapor peak for O-D stretch used to quantify the D2O vapor phase concentration is shaded in the figure; the values of pv0 are the stagnation pressures for the vapor.
Determining the microstructure of the D2O-n-nonane binary droplets is both challenging and the subject of the next chapter and will not be discussed in detail here. To summarize briefly, given the large miscibility gap exhibited by bulk n-nonane and D2O mixtures, binary droplets containing these species are unlikely to be well-mixed. Analysis of the
SAXS spectra also shows that these droplets are not phase separated, spherically symmetric structures. Molecular Dynamics simulations suggest that the droplet structure is that of a lens of nonane wetting a sphere of D2O (Pathak et al AIP Conference
Proceedings 2013). Despite these complexities, our analysis suggests that the values for
~ the overall size of the droplets, r , and number density N derived from fitting the
SAXS spectra are relatively insensitive to droplet structure. Furthermore, the small
90 difference in the scattering length density of D2O and n-nonane ensures that even reasonably large inaccuracies in composition do not significantly change the number density. In the most extreme case, i.e. treating a pure nonane aerosol as if it were pure
D2O, or vice versa, only changes the number density by ~50%. Overall, we estimate that ~ the uncertainties in r and N are less than 15%. Thus, all SAXS spectra in this chapter and Appendix D are analyzed assuming well-mixed spherical droplets.
Figure 16 illustrates the experimental results when n-nonane (pv0= 487 Pa) and D2O (pv0=
538 Pa) co-condense, together with the data corresponding to condensation of the pure components at comparable values of pv0. This is a typical case of co-condensation for these two species where particle formation is dominated by D2O nucleation but growth is dominated by n-nonane condensation. For clarity Figure 16(a) only illustrates the temperature traces. Here the dashed lines correspond to the expected isentropic values, the black lines correspond to binary condensation and the grey lines correspond to unary condensation of the two species at comparable values of pv0. Although co-condensation of n-nonane and D2O occurs ~ 15 s later than unary condensation of D2O, in both cases
TJmax = 212 K. In contrast, at the same pv0,nonane, pure n-nonane does not condense until T
= 205 K. The value of for the mixture dictates that the isentrope for binary condensation follows that for pure n-nonane. Figure 16(b) shows the average radii (black symbols) and the spread (white symbols) when n-nonane and D2O co-condense. The grey symbols illustrate the corresponding values of r for unary condensation. In the binary condensation case, r reaches a maximum value of ~13 nm as opposed to ~20 nm for
91 pure n-nonane and ~ 4.5 nm for pure D2O. Figure 16(c) summarizes the condensate mass fractions for each species. The black symbols depict gFTIR derived from the measurement of the vapor concentration, cnonane,v or cD2O,v, and the white symbols represent gFTIR determined from cnonane,l. We do not show the mass fractions obtained from SAXS, gSAXS because these values are sensitive to the assumed droplet structure. The black lines that go through the data are fits to the measurements, gfit,nonane and gfit,D2O. The dashed-dot line is gmax. The grey lines indicate the value of g during unary condensation. The n-nonane condensate mass fractions determined from cnonane,v and cnonane,l agree very well even though mass balance was not imposed as part of the FTIR spectra fitting procedure.
Although pure n-nonane condenses much further downstream compared to the binary case, near the nozzle exit the amount of n-nonane condensed is similar for the unary and binary cases. In contrast, the amount of D2O condensed based on the FTIR measurements, is much less than the maximum value, gmax and only reaches about 45% of the values observed when pure D2O vapor condenses. Thus, the presence of n-nonane adversely affects the condensation of D2O, greatly slowing the process, but in this example not cutting it off completely.
The marked decrease in the D2O condensation directly reflects changes in the flow ~ conditions –T, pv – as well as changes in droplet or aerosol characteristics – r , N , droplet temperature Td, and structure – between binary condensation and pure D2O condensation. We address all factors except structure by adapting the formalism described in detail in our recent paper on droplet growth of the pure components
(Harshad AS&T). In particular we use the rates of condensation dgfit/dt of nonane and
92 ~ D2O and N to calculate the heat flux per droplet and subsequently Td (Pathak et al 2013).
For condensation and evaporation coefficients both equal to 1, these calculations show that Td during binary condensation is ~10 K higher than during the corresponding unary nonane condensation and ~5 K higher than during corresponding unary D2O condensation experiments. Nevertheless, the condensation rate for nonane predicted using this temperature is within a factor of 2 of the observed condensation rate. This result is consistent with our previous observations (Pathak) that at the high superstations present in our experiments nonane condensation rates are insensitive to droplet temperature. In contrast, the net condensation rate of D2O after accounting for changes between the unary and binary condensation experiments is predicted to be ~15 to 20 times higher than the observed D2O condensation rate, dgfit/dt. To predict dg/dt within a factor of 2 of the observed growth rate dgfit/dt for D2O requires reducing the condensation coefficient to qc=0.45 for an evaporation coefficient qe=1 or reducing both coefficients to qc=qe=0.06-0.1.
The reduced condensation coefficients are consistent with the molecular dynamics simulations of Chakraborty and Zachariah (2008 JPC A) where the sticking probabilities of water molecules onto a water droplet coated with fatty acids (radius~4 nm) was in the range of 0.11-0.16. Similarly Takahama and Russel (Takahama and Russel 2011) found the mass accommodation of water vapor was reduced from 1 to less than 0.04 when a slab of water was coated almost fully with myristic acid- a long-chain fatty acid. They concluded that the primary mechanism for this reduction was scattering of water molecules from the hydrocarbon tails.
93 The D2O-nonane system examined here differs from the studies of water and long chain fatty acids, where bulk samples also exhibit a large miscibility gap, because an alkane molecule does not have a hydrophilic region. One significant consequence of this difference is that rather than developing the core-shell structures favored by the amphiphilic compounds (Wyslouzil et al, 2006; Li and Wilemski 2006) molecular dynamics simulations suggest D2O-water nanodroplets favor a lens-on-sphere structure
(Pathak et al, AIP Conference Proceedings 2013). Thus, the higher condensation coefficients estimated here are consistent with a droplet structure that exposes at least some regions that favor water condensation. These values of qc = qe = 0.05 – 0.10 are quite consistent with preliminary droplet structure calculations (droplet structure paper
Pathak et al) that suggest that 6-7% of the surface area of the droplet is exposed D2O, while the remainder is pure nonane.
~ Figure 16(d) illustrates the number densities N measured from SAXS experiments (black symbols) and the scaled number densities calculated by integrating the rates obtained ~ from CNT where the latter is scaled to match the maximum N SAXS . The error bars of 13%
~ on N SAXS represent an uncertainty based on the droplet structure and composition. For
~ comparison the values of N SAXS for pure n-nonane and pure D2O are depicted as grey
~ symbols. We can see that for the binary case the N SAXS values lie between those for pure
D2O and pure n-nonane, and there is some coagulation with number densities decreasing by about 20% from the maximum value of 11019 kg-1. The error bars for the pure 94 ~ droplets are less than the symbol size.. N SAXS for pure D2O droplets decreases from
19 -1 19 -1 ~ 310 kg to 210 kg , i.e. about 30%, whereas N SAXS for pure n-nonane droplets remains constant at 21018 kg-1 after the nucleation pulse. The coagulation rate for the binary condensation case is about four times than that predicted by the equation used by
Hidy and Brock (Hidy and Brock 1970, Pruppacher and Klett 1997). The predicted coagulation rate can vary by at most 20% depending on the droplet composition ranging from pure nonane to pure D2O.
Similar analysis was done for all binary traces and the results have been shown in
Appendix D.
95 D2O pv0=538 Pa and nonane pv0=487 Pa
Figure 16 Experimental parameters when n-nonane (pv0= 487 Pa) and D2O (pv0= 538 Pa) both condense together and their comparison to cases where pure vapor condenses (pv0=520 Pa for D2O and pv0=489 Pa for n-nonane). a) The temperature profile for the binary case (black line) and its comparison to the case where pure vapor condenses (grey lines). The isentrope is depicted by a dashed line. b) The average radii and the spread of the size distribution compared to the pure n-nonane and pure D2O droplets (grey symbols). c) mass fraction condensed as predicted from FTIR, gFTIR
(black symbols for fitting using the vapor spectrum and white symbols for fitting using the liquid spectrum). The black line which follows the symbols is gfit. The grey lines are gfit for cases where pure components condense. The dashed- ~ ~ ~ dot line is gmax. d) N SAXS and N scaledCNT compared with N SAXS for cases where pure components condense.
96 4.2.3 Effect of Nonane on D2O nucleation
In most of our experiments pure D2O nucleated at higher temperatures than pure nonane
(Table 3). Thus, our analysis focuses primarily on the effect the presence of nonane has on the water (D2O) nucleation rate. In the current literature, the nucleation process initiating co-condensation of water and nonane is considered to be the superposition of two unary process i.e water and nonane nucleating independently (Peeters et al. 2001;
Wagner and Strey 2001; Viisanen and Strey 1996). Our experimental results are consistent with this picture in the sense that the temperatures when co-condensation starts are never significantly higher than the temperature required to initiate condensation of one of the pure species. Furthermore, when D2O nucleation initiates co-condensation,
Figure 17(a) shows that the maximum scaled supersaturation S/Sunary for D2O reached during binary condensation is close to 1. Here, Sunary is the saturation ratio of D2O corresponding to the maximum nucleation rate for unary D2O condensation at the same pv0 (683 Pa) of D2O. In contrast, the highest S reached by nonane is only 45% of that required for pure nonane vapor to condense at the same temperature, i.e. T = 216 K.
Under these conditions Hale’s scaled nucleation model (Hale 1986, 1988 and 1992) suggests that the nonane nucleation rate is more than 4 orders of magnitude lower than the D2O nucleation rate, and, thus, nonane does not contribute to particle formation.
97 10 pv0= 325 Pa Nonane
+704 Pa D2O
TJmax=216 K
D2O 1 nonane unary S / S
0.1
20 40 60 80 100 time (s)
Figure 17 S/Sunary for D2O and nonane where D2O (pv0=704 Pa) and nonane (pv0=319 Pa) both are condensing and D2O dominates nucleation. TJmax is 216 K in this case.
We then used the D2O supersaturation profiles and the temperature profiles to calculate the nucleation pulses and characteristic times corresponding to D2O nucleation alone, as the amount of nonane is increased. Figure 18(a) illustrates the results for pv0 = 700 Pa
D2O and the three levels of nonane investigated. On one hand we observe that as we add nonane, the nucleation pulse starts earlier than in the pure D2O case. This is because the addition of nonane decreases the mixture thereby changing the relationship between p and T during the expansion. On the other hand we also see that after the peak nucleation rate is reached the pulse during binary condensation decreases more rapidly than during unary condensation of D2O. In this self quenched nucleation process, the rapid decrease in J reflects the decrease in S brought on by both the increase in temperature due
98 condensation and vapor depletion. When nonane co-condenses with water, heat is released to the flow more rapidly decreasing S more quickly and, thereby, causing J to drop more precipitously compared to the unary case. Overall, however, the increase in J prior to Jmax compensates for the decrease in J after Jmax with the net result that the characteristic time remains almost constant. For the cases illustrated in Fig 18(a), t
~11±0.4 µs, and the changes in Δt for the different binary cases are too small to establish any other trend. Earlier researchers (Wagner and Strey 2001) suggested that the additional heat released due to nonane growth cut off nucleation of water prematurely resulting in lower nucleation rates. In our experiments we find that the change in the expansion shape compensates for the more rapid quenching of the nucleation pulse and that the characteristic time is remarkably constant. If the only role played by nonane were to modify the nucleation of pure D2O in this way, we would expect the number of droplets formed to remain constant.
Figure 18(b) summarizes the particle number densities determined from SAXS for unary
D2O condensation and nonane-D2O co-condensation for the same conditions discussed in
Figure 18(a). Since number densities are calculated from SAXS spectra assuming well- mixed droplets of nonane and D2O we have included error bars to indicate the uncertainty ~ in N SAXS based on the droplet structure and composition. Contrary to our
~ expectations, N SAXS decreases systematically because as the amount of nonane increases, it decreases by up to a factor of 6 relative to the pure D2O case.
99 Figure 18 Condensation experiments for pv0=700 Pa D2O and varying amounts of nonane. (a) The nucleation pulses and characteristic times estimated based on CNT for pure D2O nucleation during unary condensation of D2O and during D2O-nonane co-condensation. (b) The measured number densities.
~ Using the values of N SAXS and Δt, we calculated the nucleation rates for D2O using
Equation (4.1) for all of the experiments where particle formation is driven by water.
Figure 19 illustrates the change in Jmax with pv0,nonane or the saturation ratio of nonane corresponding to the conditions when the D2O nucleation rate is maximized. In either picture the D2O nucleation decreases as we add nonane even though SD2O and TJmax remain essentially constant. The data for the number densities, characteristic nucleation times, maximum nucleation rates and the saturation ratios of D2O and nonane at the maximum nucleation rates of all the binary traces is given in Table D1 of Appendix D.
For the highest partial pressure of D2O Jmax decreases by about a factor of 4 between pure
100 D2O nucleation and the highest concentration of nonane. Similar trends are observed for the other two pv0 of D2O. For the lowest pv0 of D2O and the highest pv0 of nonane, nonane controls nucleation rather than D2O and this point is therefore not included in this figure.
We also plot Jmax in terms of saturation of D2O, SD2O and nonane, Snonane at Jmax. The case where lowest pv0 of D2O and intermediate pv0 of nonane condense is competitive nucleation where SD2O is reduced from 188 to 163. This is a case where Jnonane and JD2O are comparable and there are two paths for nucleation. Hale’s scaling model predicts that under these conditions nucleation rate of D2O should be ~30 times that of nonane.
Figure 19 (a) The nucleation rates of D2O as a function of the partial pressure of nonane pv at the conditions corresponding Jmax for D2O. The crosses show the predictions from Feder et al’s non-isothermal nucleation theory. (b)
The nucleation rate of D2O as a function of Snonane at the conditions corresponding to Jmax for D2O. The case when pv0=360 Pa D2O + 490 Pa nonane is not shown here because SD2O is reduced from 188 to 163 due to competitive nucleation.
101 The observation that nonane decreases Jmax at a fixed values of T and SD2O is consistent with the observations of Wagner & Strey, 1990 and Viisanen & Strey, 1993. In their NPC experiments conducted at T=230 K (Wagner and Strey 2001) and 240 K (Viisanen and
Strey 1996), they found that the activity of water required to maintain a fixed nucleation rate increased in the presence of nonane. These results differ from partially miscible systems of n-alcohols and water studied by Strey and co-workers (Strey et al. 1995) where they observed that presence of another component decreases the S for the same nucleation rate. CNT calculations – even in strong phase separating systems – always predicts that the nucleation rate for a binary system should be greater than or equal to that of the highest unary rate.
Our observations suggest that nonane is inhibiting the nucleation process of D2O in a way that isothermal CNT does not consider. The nucleation rate is related to the number of clusters of D2O that grow big enough to reach the critical size, and in CNT the critical size is a function of T and SD2O alone. In our experiments, these variables do not change significantly as we add nonane. Thus, the critical cluster size and concentration should remain the same in the presence of nonane. The decrease in nucleation rate with continued addition of nonane suggests that the second condensable species is decreasing the critical and/or subcritical cluster concentration. We hypothesize that one way nonane could do this is by reducing the cluster stability by reducing the cooling rate of the clusters by impingement with the carrier gas molecules. Feder’s theory of non-isothermal nucleation (Feder et al. 1966) suggests that the heat released due to formation of clusters of molecules decreases the nucleation rate due to heat transfer limitations. They derived
102 the following simple expression for the non-isothermal nucleation rate Jnoniso relative to the iso-thermal nucleation rate Jiso
b 2 J J (4.3) noniso b 2 q 2 iso where
2 2 N2 2 b (Cv 0.5 kB )kBT (CvN2 0.5 kB )kBT (4.4) con
As q L mv 0.5 kBT (4.5) n nn*
Here Cv and Cv-N2 are the specific heat capacities at constant volume of the vapor and nitrogen molecules respectively, kB is the Boltzmann’s constant, L is the heat of vaporization, mv is the molecular weight of the vapor and ζ is the surface tension of cluster of n molecules with a surface area of As where the critical cluster has n*
molecules. The ratio of the impingement of carrier gas and condensible vapor is N2 con and b2 represents an energy sink, in the form of carrier gas molecules, that removes the heat released due to condensation represented by q. For dilute vapors, the second term in
Equation (4.4) is much larger than the first term. All the quantities in Equation (4.4) and
Equation (4.5) are independent of the cluster size except ζAs. For D2O under the conditions typically found in our supersonic expansions, classical nucleation theory and experiments (Tanimura et al 2010) predict a cluster size of around 6-10 molecules, and
103 for a D2O monomer impinging on a cluster of 8 molecules, we find that the change in surface energy is ~ 4 kBT assuming a spherical cluster. This value is less as compared to
L mv of 27 kBT. When a nonane monomer impinges the critical cluster, it is difficult to
exactly determine ζAs and we assume that it is negligible as compared to L mv of 30
kBT. Thus q L mv in both cases and nonane and D2O have similar values for q. We
assume that q 25k BT when nonane is also a possible condensate along with D2O.
However, con changes due to the presence of an additional condensible species. The net effect is that b decreases, q remains constant and Jnoniso decreases as the amount of nonane increases. The crosses in figure 19(a) illustrate the calculation for the highest pv0 of D2O (SD2O=101+4) . We see that our calculations for Jmax decrease with an increase in
Snonane which is a consistent trend with our observations suggesting that non-isothermal nucleation may be one of the reasons for this phenomenon. We also investigated the pressure effect on our nucleation rates (Wedekind et al 2008) and found that incorporating the pressure effect would decrease them by less than 0.1%.
104 D2O pv0=367 Pa and nonane pv0=618 Pa
Figure 20 The experimental parameters when both nonane (pv0= 618 Pa) and D2O (pv0= 367 Pa) are condensing together compared to the conditions when pure nonane (pv0=625 Pa) and pure D2O (pv0=346 Pa) are condensing. The symbols are explained in Figure 16. This is the only case where nonane dominates nucleation and after the onset of nucleation, there is negligible condensation of D2O as seen from the FTIR result in figure c. In figure d, the nucleation pulse is calculated for nonane.
We also examined the one case where nonane appears to control nucleation. This case, illustrated in Figure 20, corresponds to pv0= 618 Pa nonane + 367 Pa D2O. Fig 10 (a) 105 illustrates the temperature profile for the binary case as well as the corresponding unary cases of nonane (pv0=625 Pa) and D2O (pv0=346 Pa). The temperature profiles shown here are those corresponding to the PTM alone TPTM, rather than on the integrated analysis, because for this experiment it was difficult to combine gPTM and gFTIR accurately near the onset or condensation. Fortunately, for the purposes of the current analysis, near the onset of condensation TPTM is accurate enough. Figure 20(a) illustrates that during binary condensation the temperature profile follows that of pure nonane at the comparable pv0 and has the same TJmax as pure nonane. Figure 20(b) shows that in the presence of D2O the final droplet size decreases to
D2O in the droplet and we can assume that the droplet is made of pure nonane with a spherical structure. Finally Figure 20(d) summarizes the number densities. Here, the ~ results demonstrate that in the presence of D2O the final number density, N SAXS = 2.2
18 -3 ~ 18 -3 10 cm , is higher than that that observed for pure nonane, N SAXS = 1.4 10 cm , consistent with the observed decrease in droplet size in the presence of D2O. The characteristic nucleation times, based on the saturation profile for nonane, Figure 21, are
17 μs for the binary case and 16 μs for pure nonane at pv0=625 Pa. The nucleation rate in the binary case is J=1.5 1016cm-3s-1 , a value that is ~ 50% higher than that observed for
15 -3 -1 the nucleation rate of pure nonane at a similar pv0, J = 9.410 cm s .
106 Although these nucleation rates are quite close to each other, the difference in Jmax cannot ~ be attributed to an uncertainty in N SAXS alone. As illustrated in Figure 21, although the
S/Sunary curve for nonane again reaches value very close to 1 and always lies above the corresponding S/Sunary curve for D2O, the maximum for the latter is ~0.68. Here, Sunary for
D2O at T = 210 K is determined by interpolating the Sunary vs TJmax data for pure D2O condensation.
10
pv0= 618 Pa Nonane
+367 Pa D2O
TJmax=210 K
1 unary S / S
D2O nonane 0.1
60 80 100 120 time (s)
Figure 21 S/Sunary for the case where nonane appears to dominates nucleation(TJmax =210 K). D2O (pv0 = 367 Pa) and nonane (pv0 = 618 Pa).
For the conditions in Figs. 20 and 21, Hale’s scaling model (Hale 1986,1988 and 1992) suggests that the pure component nucleation rates should be within a factor of two of each other, with a maximum nonane nucleation rate of Jmax(S=1161, TJmax=210.4 K) =
15 -3 -1 6.110 cm s and a maximum D2O nucleation rate of Jmax(S=94, TJmax=210.4 K) = 1.2
107 1016 cm-3s-1. Here the scaled model prediction for the water nucleation rate does not account for any reduction in Jmax due to the presence of nonane, and effect that should be significant. Furthermore, these calculations are quite sensitive to small changes in temperature e.g. an increase in temperature by 0.8 K can cause the nucleation rate to decrease by 50% . Nevertheless, the scaling model supports the idea that although D2O does not initiate nucleation under conditions illustrated in Figure 10, the D2O nucleation rates under these conditions are comparable to those for nonane and thus can explain the increase in the nucleation rates when D2O is present relative to the nucleation of nonane alone.
Additional experiments at higher alkane partial pressures and lower water partial pressures are required to determine whether D2O can also reduce the rate of nonane nucleation in a manner consistent with the results of Wagner and Strey (1990 and 2001) and with the effect of nonane on D2O nucleation observed in the current work.
4.3 Summary and Conclusions
We investigated the nucleation and condensation behavior of supersaturated vapors of nonane and D2O in a supersonic nozzle. We performed an extensive set of experiments covering of three nominal pv0 of nonane, three nominal pv0 of D2O and the nine binary combinations. We find that under our experimental conditions, in seven of the nine binary condensation cases D2O nucleation dominates particle formation. In these cases,
D2O initiates nucleation but the presence of nonane reduces the total number of droplets formed and also inhibits of the further condensation of D2O. The reason for fewer
108 droplets can be attributed at least in part to non-isothermal nucleation theory that predicts a decrease in the nucleation rate in the presence of a second condensing species, although quantitative agreement is still lacking. This result is similar to that observed by Wagner and Strey (2001 and 1990) where they observed an increase in SD2O required to maintain the same nucleation rate in the presence of nonane. The subsequent inhibition of D2O condensation can be attributed both to an increase in the temperature of the growing droplets in the presence of nonane and to droplet structures where most of the surface is comprised of hydrophobic nonane molecules.
In one of the nine binary cases nonane appears to dominate nucleation. Although at the onset of condensation the Smax value for nonane is similar to that for pure nonane at similar pv0, the nucleation rate of D2O predicted under these conditions from Hale’s scaling model is comparable to that predicted for nonane. As a result the nucleation rate when D2O is present is ~50% higher than the nucleation rate when pure nonane vapors at similar pv0 are condensing.
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113 Chapter 5 The Structure of D2O-Nonane Nanodroplets
This chapter is the basis for a manuscript titled “The Structure of D2O-Nonane
Nanodroplets” that will be submitted for publication. The authors for this manuscript are
Harshad Pathak, Barbara Wyslouzil, Abdalla Obeidat and Gerald Wilemski. The data presented in figures 26, 27 and 28 in this chapter were first published in AIP conference proceedings, Volume 157, page 472-475 (2013). The experiments discussed in this chapter were performed by Harshad Pathak and Barbara Wyslouzil. The molecular simulations were performed by Gerald Wilemski and Abdalla Obeidat. Dr. Wilemski also derived the analytical expressions for the form factor, radius of gyration rG and the scale factor I(0). Harshad Pathak completed the analysis of the experimental results, wrote the code to fit the experimental data to the new form factor incorporating the effects of aerosol polydispersity, and also performed the Guinier analysis.
114 5.1 Introduction
Aqueous-organic nanodroplets are a subject of continuing interest for environmental and aerosol scientists (Charlson et al. 2001). Some of these nanodroplets can grow large enough to form Aitken nuclei and possibly, cloud condensation nuclei. The microstructure of these droplets is an important parameter that determines their chemical properties and their albedo (Rudich 2003; Eliason et al. 2004). For droplets small enough to act as critical nuclei in vapor-liquid phase transitions, surface composition determines the surface free energy which in turn strongly affects the rate of formation of these droplets (Charlson et al. 2001; Chen et al. 2003; Facchini et al. 1999).
The structure of aqueous-organic droplets depends both on the bulk solubilities of the species and whether any region of one molecule interacts favorably with the others. Many organic compounds have very low solubilities in water, and, as a result, these mixtures display miscibility gaps separating water-rich and organic rich phases within the droplet
(Li and Wilemski, PCCP 2006, Nellas et al., 2007; McKenzie and Chen, 2006). In particular, surface active components are likely to form films on the outer surface of droplets (Ellison et al. 1999, Wyslouzil et al 2006), and indeed Tervahattu et al. found eveidence that fatty acids form a surface coating over marine aerosols (Tervahattu et al.
2005; Tervahattu et al. 2002). Aqueous-organic aerosol droplets may also adopt radically different structures depending on their overall composition (Li and Wilemski 2006), and, as discussed in more detail in this chapter, may not necessarily be spherical.
The goal of this work is to probe the internal structure of binary aqueous-organic nanodroplets by combining the results of experiments and molecular dynamics (MD) 115 simulations. We produce the binary nanodroplets by co-condensing D2O and nonane in a supersonic nozzle, and measure the Small Angle X-Ray Scattering (SAXS) spectra of the composite nanodroplets. To estimate the average size of these droplets, we first assume a structure for the droplet and derive its scattering form factor, and then analyze the SAXS spectra so that the theoretical calculation, after incorporating polydispersity, matches the experimental results. We narrow down the possible structures of these nanodroplets based on the results of Density Functional Theory (DFT) calculations and Molecular Dynamics
(MD) simulations. Although the x-ray scattering contrast between nonane and D2O makes it difficult to distinguish between these structures, we can use the fit parameters to determine the overall composition of the aerosol and compare this to independent composition measurements from Fourier-transformed Infrared Spectroscopy (FTIR). We find that a D2O core-nonane shell structure is inconsistent with the results of MD simulations and SAXS even though water and nonane are immiscible and D2O has a surface tension three times higher than nonane. Instead, we find that the droplet structure most consistent with SAXS fitting and MD simulations is equivalent to a spherical lens of nonane on a sphere of D2O. Even this structure, however, under predicts the amount of
D2O condensed by around 30-40% as compared to FTIR and over predicts the amount of nonane by ~10%.
5.2 Theory
With a volume fraction on the order of 10-6, the aerosols formed in this work are dilute.
Thus, the inter-particle correlations are extremely weak and the scattered intensity I(q) as a function of the scattering vector q can be approximated by the sum of contribution of all
116 particles. For an ensemble of particles over a size range j, the scattered intensity is given by
I(q) NP(q) N j Pj (q) (5.1) j where N is the number density of particles and P(q) is the mean particle form factor. The
particle form factor for size range j, Pj (q) is defined as
2 Pj (q) f j (q) (5.2)
where f j (q) is the particle form amplitude for a particle of size j. The brackets depict the average over all possible orientations of the particle, and are required when particles do not have spherical symmetry. The particle form amplitude for a particle of size j and for a scattering vector q is defined as f (q) (r) exp iq r dr (5.3) j Δρb Vj
where b (r) is the difference in the scattering length density of a particle located at position r with respect to its surroundings. The integration is carried out over the effective particle volume Vj.
Recent molecular dynamics simulations of water-nonane nanodroplets indicate that the composite droplets are moderately nonspherical and strongly phase-separated into water- rich and nonane-rich regions (Hrahsheh and Wilemski, 2013). A simple, but realistic model of these droplets is a spherical lens of nonane that partially wets a sphere of water
117 c R1 R R2 R1 R2 1 R2 d d d
(a) (b) (c)
Figure 22 (a) An illustration of the sphere-in-sphere structure derived from DFT. There is a sphere of D2O (radius=R2) inside a sphere of nonane (radius=R1). d is the distance between the two centers. θc is the contact angle of nonane on
0 D2O and is 0 in this case. (b) An illustration of the lens-on-sphere structure derived from MD simulations. θc is 76 in this case. (c) An illustration of the sphere-outside-sphere structure derived from simplifying the MD simulations
0 results. θc is 180 in this case.
with a contact angle θc as shown in Figure 22. For this model, the integration to find fj(q) is performed using a method similar to that of Fütterer, Vliegenthart, and Lang (Fütterer et al. 2004) who treated the special case of a hemispherical cap of uniform density. The final expression (Wilemski et al., 2013) for Pj(q) is given in Appendix E.
For different values of θc, the lens-on-sphere model reduces to the special cases of
o o sphere-on-sphere (θc =180 ) and sphere-in-sphere (θc ≈ 0 ) models. The sphere-in-sphere structure was suggested by earlier density functional theory (DFT) calculations for a binary hard sphere-Yukawa system (Hrahsheh and Wilemski, 2013). These model structures are illustrated in Figure 22. After calculating the form factors for these structures from scattering theory, we incorporate polydispersity by deriving the polydisperse form factor Pff (q) numerically-
118 r4 Pff (q) P (q) pdf dj (5.4) j 0 where pdf is a probability density function for the number distribution of particles. The upper limit of the above integral should be infinity but for our experiments we find that for values of j greater than r + 4σ, the integrand is almost zero. The probability density function which we use is a Schulz function
z1 z 1 z 1 pdf ( j) j z exp j / (z 1) (5.5) r r where z=(σ/ r )2-1. The theoretical scattering intensity is then calculated by
I(q) Pff (q) N (5.6)
5.3 Experiments and their analysis
The experimental set-up and techniques used in the current work are described in detail in
Chapter 2. The experimental conditions noted in Table 4 are the same as those summarized in Table 3 in Chapter 4. Table 3 lists only the total amount of condensate, whereas Table 4 includes both the mass fraction of nonane and and the mass fraction of
D2O entering the nozzle.
119 Table 4 gmax and TJmax as a function of the nominal pv0 for D2O and nonane. The actual pv0 are within 20 Pa of these values and the exact value corresponding to each experiment is listed in Table 6. TJmax is the temperature corresponding to the maximum nucleation rate and gmax is the mass fraction of the condensible vapor entering the nozzle.
Nonane pv0=320 Pa pv0=490 Pa pv0=620 Pa
D2O TJmax(K) gmax,D2O gmax,nonane TJmax(K) gmax,D2O gmax,nonane TJmax(K) gmax,D2O gmax,nonane pv0=360 Pa 205 0.0082 0.047 206 0.0082 0.070 210 0.0081 0.088 pv0=540 Pa 212 0.0124 0.046 212 0.0121 0.070 213 0.0121 0.088 pv0=700 Pa 216 0.0162 0.048 216 0.0161 0.070 216 0.0159 0.088
5.3.1 Analysis of SAXS spectra for unary droplets
Figure 23 shows representative fits to the pure component SAXS spectra assuming that scattering arises from a polydisperse collection of spheres that follow a Schulz distribution. The theory captures the experimental data quite well. The scattered points are the experimental data and the lines are the fits. The region of low q, i.e. where q r <1, is called as the Guinier region, whereas the region where q r ~1 is called the
Porod region. The kink in the data points of figure 23(a) indicates that the aerosol is
-1 relatively monodisperse and the location of the kink at ~0.35 nm when pv0=322 Pa, indicates r where q r ≈3π/2. Thus
2 2 I(0) N V (b ) (5.7)
120 Figure 23 A typical fitting case where we fit the pure component spectra to theoretical scattering from polydisperse spheres. The value of r are obtained from the fitting parameters. (a) Nonane spectra are shown when droplets are at the nozzle exit. The spectra for r =25 nm and r =20 nm have been offset by 100 and 10 respectively for clarity.
(b) D2O spectra are shown when droplets are at the nozzle exit. The spectra for r =5.1 nm and r =4.6 nm have been offset by 100 and 10 respectively for clarity.
where V 2 is the average of the square of the volume of a particle incorporating polydispersity.
Fitting of the SAXS spectrum using the non-linear optimization program provided by the
Advanced Photon Source (APS) (Kline, 2006) gives r and N. The data for pure nonane droplets begin at a low q of 0.08 nm-1 because the droplet sizes are relatively large as
-1 opposed to pure D2O, where the data start at q~0.2 nm because these droplets are smaller. This is because we changed the sample-to-detector distance for the D2O 121 experiments to 0.85 m from 2.13 m for nonane to probe smaller droplets and the data at lower q would not be useful to us. We see that for pure compounds, the experimental data fit the assumption of scattering from homogenous polydisperse spherical drops.
To quantify the goodness-of-fit we plot the reduced chi-squared value of the fits as a function of position in the nozzle. The expression for this is as follows
Npo int s 2 2 1 (Oi Ei ) X N po int s (5.8) N po int s i1 Ei
where Npoints is the number of data points being fitted, Oi is the observed value of the data point i and Ei is the expected value from the fitted model. This graph is shown in Figure
24. In general, a low value of reduced chi-squared indicates that the fit is good. Figure 24 shows that when our data fit the theory well, the reduced chi-squared values are less than
10. We observe a broad trend of an increase in the reduced chi-squared value with respect to position from the throat because as we move downstream of the nozzle, droplet number densities increase which results in larger values of intensity I(q) for same values of q. Thus, we have larger values of Oi and hence larger absolute deviations, Oi-Ei from the fitted model.
122 14 nonane pv0=625 Pa 12 ) D2O pv0=683 Pa 10 points
/N 8 2 6
4 sqrt (X sqrt
2
0 3.5 4.0 4.5 5.0 5.5 6.0 6.5 7.0 7.5 distance from throat (cm)
Figure 24 The reduced chi-squared values of SAXS fitting as we move downstream of the nozzle.
5.3.2 Analysis of Binary SAXS spectra
To start, we analyze the binary spectra assuming the droplets are well-mixed spheres.
Although we know that water and nonane do not mix macroscopically, we want to investigate if the SAXS are consistent with well-mixed spherical structures when the droplet sizes are of the order of nanometers. Figure 25(a) illustrates the fits to the data near the nozzle exit at three different compositions of condensing vapor, and Figure 25(b) summarizes the normalized Chi-squared values as a function of position for these three vapor compositions. In our experiments, the maximum deviation between the data and fit occurs when the concentration of nonane in the condensing vapor is decreased relative to the amount of D2O. As in the unary case, chi-squared tends to increase as we move downstream of the nozzle and droplet size increases. Here the trend in the increase in the reduced chi-squared values is clearer than in Figure 24 because for a given droplet
123 composition, the mismatch between experimental data and predicted values from the model is roughly proportional to the droplet size. In the worst cases, chi-squared is more than 2.5 times higher than in the fits to the pure component aerosol. The largest deviations between the data and the fit occur for q > 0.3 nm-1. Although the degree of deviation differs between cases, the nature of the deviation is the same for all of the binary spectra. We find that the model of well-mixed spheres does not predict our experimental data well and hence, we look at scattering from other particle shapes. To enhance our ability to understand droplet structure, we focused on the spectra corresponding to the highest pv0 of D2O and lowest pv0 of nonane at the nozzle exit
1e+2 p = 618 Pa nonane v0 (a) 40 nonane D O and 367 Pa D O 2 (b) 1e+1 2 pv0 (Pa) pv0(Pa) p = 487 Pa nonane v0 325 704 ) 1e+0 and 538 Pa D2O 30 487 538 p = 325 Pa nonane 1e-1 v0 618 367 and 704 Pa D O points -1 2 /N
1e-2 2 20 /cm I 1e-3 1e-4 sqrt (X sqrt 10 1e-5
1e-6 0 0.01 0.1 1 0 2 4 6 8 -1 q (nm ) distance from throat (cm)
Figure 25 (a) The fits of binary SAXS spectra assuming the aerosol is a polydisperse collection of well-mixed spheres.
The spectra for 618 Pa nonane + 367 Pa D2O and 487 Pa nonane + 538 Pa D2O have been offset by factors of 100 and
10 respectively for clarity. (b) the reduced chi-squared plotted as a function of distance from the throat for the cases mentioned in (a). The worst fits correspond to droplets that form when the D2O concentration high and the nonane concentration is low.
124 because this is the spectrum that exhibits the largest deviation from the scattering from well-mixed droplets. The structure we determine by fitting this scattering spectrum is expected to be representative of the structure for all binary droplets.
Next, we considered two forms of the axisymmetric core-shell structure. These are illustrated in Figure 26 and the values in the inset of the graph are the fit parameters.
Here, is the volume fraction of the aerosol, σ is the spread of the size distribution, bkgd is the background intensity, r core is the average core radius and δshell is the shell thickness.
The most physically reasonable case is that where D2O forms the core of the droplet while nonane, with a surface tension that is ~30% that of D2O, forms the shell of the droplet. When, however, we analyze our experimental data using this assumption, the resultant fit is identical to that derived for the well-mixed droplet. The fit converges to either a pure D2O droplet or a pure nonane droplet with the same values of r , sigma σ and I0. The values of the aerosol number density, or volume fraction differ by the
2 2 , 9.13104 nm 2 factor of b,D2O = 1.49 where and are the 4 2 b,D2O b,nonane b,nonane 7.4810 nm difference between the scattering length density of pure D2O or nonane particles with
respect to surrounding N2 gas. The values of b,D2O and b,nonane used in this analysis are kept constant at 9.13104 nm2 and 7.48104 nm2 respectively which are calculated
125 for a temperature of 235 K. Incorporating the change in b with respect to temperature changes our r and σ by less than 1%.
Surprisingly, when we switch the locations of the D2O and nonane within the drop, the fit to the data appears quite good with reduced chi-squared values of less than 10.
Unfortunately, this structure is unrealistic for a number of reasons: the large difference in the surface tension between the two compounds means that this is not a thermodynamically stable configuration, and furthermore, water does not easily wet nonane and a core shell structure implies a contact angle of 0°. Finally, the fit parameters for this model predict a D2O condensate mass fraction that is 2.8 times the amount of
D2O entering the nozzle, severely violating mass balance.
1e-1 1e-1 (a) (b)
1e-2 N 1e-2 D D N
1e-3
1e-3 -1 -1 /cm /cm 1e-4 1e-4 = 6.27E-6 I = 4.59E-6 I
Figure 26 Fitting of the SAXS spectra to (a) a D2O core-nonane shell structure and (b) nonane core-D2O shell structure.
126 Table 5 summarizes the fit parameters derived assuming well-mixed or core-shell droplet structures. In addition to r , N, and polydispersity ff=σ/ r , the table also reports the fraction of D2O condensed, fraction of nonane condensed, the volume fraction of the aerosol and the average weight% D2O in a droplet. For the well-mixed droplet assumption, the composition of the droplets is based on the FTIR analysis that found 66% of the D2O and 84% of nonane condensed, and, thus, the droplet is 21wt% D2O. When we assume a D2O core-nonane shell structure, the fit parameters are identical to the well mixed case since the best fit always corresponded to r = 0 depending on core = 0 or δshell the initial parameters estimates. In contrast, the nonane core-D2O shell yields fit parameters where the droplet composition is 88wt% D2O and the D2O mass fraction condensate violates mass balance.
127 Table 5 The sensitivity of fit parameters to different droplet structures. The numbers are bolded when mass balance is violated.
-3 6 Droplet r N (m ) ff gD2O/gmax gnonane/gmax 10 Wt.% structure of
D2O in drop Well-mixed 10.62 9.2 1017 0.29 0.66 0.84 5.83 21
17 D2O core- 9.9 10 0.29 0 (pure 1.09 (pure 6.27 0% or r core=0 or 10.62 Nonane shell (pure nonane) nonane) or 100% or δshell = 10.62 or 0 nonane) or 3.13 0 (pure
17 (pure D2O) 4.2 r total=10.62 6.6 10 D2O) (pure D2O) Nonane 7.1 1017 0.27 2.83 0.13 4.59 88 r core=5.87 core-D2O shell δshell=4.93
r total=10.8
Bimodal distribution- i) i)12.8 & 8.5 i)4.9 1017 i)0.20 i)0.66 i)0.84 i)5.7 i)21 g =g nonane FTIR & 3.1 1017 & &g =g D2O FTIR 0.20
ii) 5.5 1017 ii) ii) 12.2 & 4.6 ii)0.83 ii)0.84 ii)5.91 ii)25 & ii)0.2 g =g nonane FTIR 10.8 1017 2 & iii) 0.63 iii)12.0 & 3.3 iii) iii)0.66 iii)0.90 iii)6.05 iii)20 g =g D2O FTIR 6.2 1017 & iii)0.2 17.0 1017 3 & 0.77 Sphere-in- 10.6 & 3.2 9.9 1017 0.30 0.12 1.06 6.25 4 sphere
Sphere-on- 11.0 & 4.4 8.9 1017 0.25 0.28 1.05 6.37 8 sphere
Lens-on- 11.3 & 4.8 8.2 1017 0.25 0.35 1.01 6.27 10 sphere
128 An alternative way to interpret the spectra is to invoke a mixed population of polydisperse spheres, for example, a population of pure nonane spheres and a second population of pure D2O spheres. Figures 27(a)-(c) illustrate the results of fitting the
SAXS spectra using this assumption. The differences between the three graphs correspond to different constraints imposed on the fit. In (a), the amounts of nonane and
D2O condensed are assumed to equal that measured by FTIR, and the fit does not agree well with the data. When we relax one of the constraints and allow the amount of D2O or nonane to vary (Figures 27(b) and 27(c)), we get better agreement between the data and the fits. In all cases, the “bimodal” fit suggests a population of ~12 nm nonane droplets mixed with a population of smaller D2O droplets, where the size and number density of the D2O droplets is a strong function of the imposed constraint.
Although bimodal fits can be reasonably consistent with the SAXS and FTIR data, we are skeptical that a bimodal distribution of droplet sizes can exist in our experiments. A bimodal distribution requires either two nucleation events, or significant droplet break-up to occur. Based on the analysis presented in Chapter 4, we only observe one nucleation pulse during co-condensation that is dominated either by the nucleation of D2O or by the nucleation of nonane. Droplet breakup is also considered to be an extremely unlikely event and has never been observed in the MD simulations. Finally, the number of adjustable parameters involved in bimodal fits is large and if not enough constraints are invoked, many parameter combinations can be used to fit the data quite well.
129 Nevertheless, based on physical arguments it is unlikely that our binary droplets exhibit a bimodal droplet size distribution.
Figure 27 Fitting of the scattering spectrum to a bimodal distribution of droplets (a) Nonane and D2O volume fractions are fixed to those derived from FTIR while fitting (b) volume fraction of nonane is fixed and that of D2O is allowed to vary(c) volume fraction of D2O is fixed and that of nonane is allowed to vary.
DFT calculations by Wilemski and co-workers (Wyslouzil et al., 2012) for binary
Yukawa hard-sphere mixtures predict a binary droplet structure that consists of a sphere of D2O inside a larger sphere of nonane. We simplify this structure assuming that the two
o spheres touch each other at their periphery (θc 0 ), calculate the form factor for such a structure using scattering theory assuming the droplets follow a Schulz distribution function with the relatively polydispersity of nonane and D2O being the same. Figure
28(a) illustrates the fit for this model and once again we see that agreement between the fit and the experimental data is not that good. Furthermore, the amount of nonane condensed predicted from this fitting is 1.06 times the amount entering the nozzle and thus, violating mass balance and ~20% above the values measured by FTIR.
Nevertheless, the fit parameters, are very similar to the first three structures.
130 MD simulations predict a lens-on-sphere model where a spherical lens of nonane partly covers a “sphere” of D2O. If this is simplified to assume that a sphere of nonane touches a
o sphere of D2O on the outside (θc 180 ), we can easily derive the form factor by modifying our earlier expression for the sphere-in-sphere case. Although figure 28(b) shows that this form factor fits the data well, only 28% of the D2O entering the nozzle condenses, a value that is rather low compared to the 66% measured by FTIR.
Furthermore, the amount of nonane condensed is also 5% higher than that entering the nozzle.
For the lens-on-sphere model, Wilemski et al.’s (Wilemski et al., 2013) MD simulations
o o o indicate θc = 76.6 at 220 K and θc = 76 at 250 K. We therefore use 76 as the contact angle and derive the form factor Pl(q) using equations (5.2) and (5.3). We then fit the scattering data to the scattering from a polydisperse collection of particles that have this structure using the APS non-linear optimization program. The best fit using this structure as illustrated in figure 28(c). The parameters of this fit predict that 35% of the D2O and almost all of the nonane is condensed.
131 Figure 28 Fitting of the scattering data to scattering from (a) a sphere inside sphere structure (b) a sphere on sphere structure (c) a lens-on-sphere structure.
We examined the effect of varying θc when fitting the data but, as illustrated in Figure 29, the results are not that sensitive to θc, except near the extremes. Indeed, the value of θc that gives the best agreement with the FTIR results for the mass fraction of D2O
o condensed is θc = 76 . On the right axis, the reduced chi-squared value is maximized
(~20) when θc is 0 suggesting this fit is the worst for the symmetric sphere-in-sphere droplet shape. The value of the reduced chi-squared decreases and stays at ~7 as θc
o increases, reaching a value of ~11 when θc is 180 . In other words, all of the structure that do not have spherical symmetry do a better job at describing the scattering data.
132 1e-6 FTIR 30
8e-7 )
N N D D N D points 6e-7 20 /N 2 D2O
4e-7 10 2e-7 Sqrt (X
0 0 0 50 100 150 Contact angle (o)
Figure 29 D2O and reduced chi-squared for different contact angles of a lens-on-sphere model.
We also checked the consistency of the full fits by fitting the data in the Guinier region, i.e. the region where q r <1. In this region
2 2 ln I(q) ln I(0) q rG / 3 (5.9)
and a straight line fit yields the values for radius of gyration, rG and I(0). For the lens on sphere, Appendix E summarizes the equations that relate I(0) and rG to the model parameters and the effective scattering length density. Figure 30 shows the plot of ln(I) vs
2 q , the Guinier plot, for the measured scattering intensity for our test case of D2O
(pv0=325 Pa) and nonane (pv0=625 Pa). In this case the low q data exhibit good linearity and are easily fit to a straight line.
133 0
-1 2 2 ln(I) = -rG /3*q - ln(I0) 2 ) -2 ln(I) = -48.29*q - 0.9442 -1
-3 /cm I
ln( -4
-5
-6 0.00 0.02 0.04 0.06 0.08 0.10 0.12 q2 (nm-2)
Figure 30 Guinier plot for the test case of D2O (pv0=325 Pa) and nonane (pv0=625 Pa). The circles show the raw data and the line shows the fit through the linear section of the data points.
The expressions for rG and I(0) are complicated for the lens-on-sphere structures and are given in Appendix E. The expression for rG depends non-linearly on Rlens, Rsphere and polydispersity whereas the expression for I(0) depends linearly on N and non-linearly on
R1, R2 and polydispersity. Both the expressions also depend on scattering length density and contact angle of the lens on the sphere which are known quantities. Using the values for R1, R2 and polydispersity derived from the full fit (Kline, 2006) and the expression for
2 rG, we find rG = 144 nm . This compares well with direct measurement of the Guinier
2 17 -3 slope which yields rG = 145 nm . The value for N from the full fit is 8.2 10 m a value that is within ~12% of the value 7.1 1017 m-3 that is derived from the Guinier intercept after using the values of R1, R2 and polydispersity from the results of the full fit.
134 These two values are within an experimental uncertainty of N and suggest that the assumption that the droplets have a lens-on-sphere structure is reasonable.
We then analyzed the SAXS spectra at the nozzle exit for each of the remaining cases of binary condensation using both the full fit and the Guinier analysis and in all cases there is good agreement between the two with respect to rG and N. Moreover, these values are close to those derived assuming the well-mixed spherical structures. These results indicate that rG and N for the binary droplets derived by analyzing the SAXS spectra are not particularly sensitive to the assumption of droplet structures for these spectra. Table 6 summarizes the results for N and rG derived using the full fit and the Guinier analysis assuming a lens-on-sphere droplets structure and compares these with N and rG derived assuming well-mixed spherical droplets. The vaues of pv0 are the actual experimental values rather than the nominal values shown in Table 4.
135 Table 6 N and rG derived using different analyses of SAXS spectra at nozzle exit for the nine case of binary condensation.
Nonane D2O Lens-on-sphere- full Lens-on-sphere-Guinier Well-mixed spheres- full fit analysis fit pv0 (Pa) pv0(Pa)
319 357 2 2 2 2 2 2 rG = 135 nm rG = 126 nm rG = 135 nm
N = 1.2 1018 m-3 N = 1.1 1018 m-3 N = 1.1 1018 m-3
311 538 2 2 2 2 2 2 rG = 138 nm rG = 132 nm rG = 134 nm
N = 1.1 1018 m-3 N = 0.93 1018 m-3 N = 1.1 1018 m-3
325 704 2 2 2 2 2 2 rG = 144 nm rG = 145 nm rG = 134 nm
N = 8.2 1017 m-3 N = 7.1 1017 m-3 N = 9.2 1017 m-3
487 364 2 2 2 2 2 2 rG = 265 nm rG = 233 nm rG = 266 nm
N = 6.4 1017 m-3 N = 5.5 1017 m-3 N = 6.3 1017 m-3
487 538 2 2 2 2 2 2 rG = 231 nm rG = 217 nm rG = 225 nm
N = 7.4 1017 m-3 N = 6.6 1017 m-3 N = 7.6 1017 m-3
489 714 2 2 2 2 2 2 rG = 233 nm rG = 217 nm rG = 222 nm
N = 7.0 1017 m-3 N = 6.0 1017 m-3 N = 7.7 1017 m-3
618 367 2 2 2 2 2 2 rG = 551 nm rG = 501 nm rG = 559 nm
N = 2.1 1017 m-3 N = 1.8 1017 m-3 N = 2.0 1017 m-3
622 547 2 2 2 2 2 2 rG = 335 nm rG = 301 nm rG = 335 nm
N = 5.4 1017 m-3 N = 4.7 1017 m-3 N = 5.2 1017 m-3
621 718 2 2 2 2 2 2 rG = 314 nm rG = 285 nm rG = 308 nm
N = 5.8 1017 m-3 N = 5.0 1017 m-3 N = 5.9 1017 m-3
136 Finally, we calculated the droplet composition obtained from the lens-on-sphere structure and compared those values to the results obtained by FTIR. Table 7 compares g/gmax for nonane and D2O and the droplet composition at the nozzle exit obtained from the lens-on- sphere structure to that derived from FTIR. For D2O g/gmax predicted by the lens-on- sphere structure is generally less than that predicted by FTIR except for the lowest pv0 values of D2O. In this case the FTIR measurement of gD2O is less than that from the fit of the SAXS spectra to the lens-on-sphere structure. For nonane the g/gmax values derived from the SAXS fitting is 10 – 20% higher than that found by FTIR. At times the amount of nonane condensed is up to 9% higher than that is entering, thereby violating mass balance. The final column is the weight percent of D2O in the droplet. In all cases more nonane condenses than D2O. Our observation is consistent with results reported by van
Dongen and co-workers (Peeters et al., 2004) where they also observed that once either nonane or D2O nucleates, the subsequent condensation is dominated by nonane.
137 Table 7 g/gmax for nonane and D2O at the nozzle exit along with the weight percent of D2O for the nine sets of experiments. FTIR results for pv0 = 618 Pa nonane and 367 Pa D2O shows very little condensation of D2O which is difficult to quantify. This is the case where nucleation and the subsequent droplet growth is dominated by nonane.
Nonane D2O g/gmax (D2O) g/gmax (nonane) Weight percent of D2O pv0 (Pa) pv0(Pa)
319 357 FTIR lens-on-sphere FTIR lens-on-sphere FTIR lens-on-sphere
0.45 0.32 0.93 1.04 7.8 5.1
311 538 FTIR lens-on-sphere FTIR lens-on-sphere FTIR lens-on-sphere
0.66 0.31 0.88 1.09 17 7.2
325 704 FTIR lens-on-sphere FTIR lens-on-sphere FTIR lens-on-sphere
0.66 0.35 0.84 1.01 21 10
487 364 FTIR lens-on-sphere FTIR lens-on-sphere FTIR lens-on-sphere
0.17 0.22 0.92 0.99 2.1 2.5
487 538 FTIR lens-on-sphere FTIR lens-on-sphere FTIR lens-on-sphere
0.41 0.31 0.88 1.01 7.4 5.0
489 714 FTIR lens-on-sphere FTIR lens-on-sphere FTIR lens-on-sphere
0.50 0.31 0.83 1.01 12 6.6
618 367 FTIR lens-on-sphere FTIR lens-on-sphere FTIR lens-on-sphere
0 0.14 0.87 0.95 0 1.3
622 547 FTIR lens-on-sphere FTIR lens-on-sphere FTIR lens-on-sphere
0.21 0.20 0.88 0.97 3.2 2.7
621 718 FTIR lens-on-sphere FTIR lens-on-sphere FTIR lens-on-sphere
0.35 0.23 0.84 0.98 7.0 4.0
138 5.4 Summary and Conclusions
By analyzing the SAXS spectra of nonane-D2O nanodroplets, we are able to gain insight into their structure. Complementary MD simulations suggest that the nanodroplets form a nonane lens-on-D2O sphere structure. Using the form-factor derived for such a structure, and incorporating polydispersity, provided a better fit to the SAXS spectra than other droplet structures like core-shell structures or sphere-inside-sphere structures. After determining the droplet structure, we also perform a Guinier analysis. The consistency between the Guinier region and the Porod analysis regarding rG and N is encouraging.
But, we should also be aware that N and rG are relatively insensitive to droplet structures.
The overall droplet composition is also measured from FTIR experiments and the amount of D2O predicted from the SAXS analysis is generally less than that measured by FTIR.
Also, amount of nonane condensed predicted by SAXS analysis is at times higher than that entering the nozzle, thereby violating mass balance. Nevertheless, the resultant aerosol compositions derived from the lens-on-sphere structure were more consistent with the FTIR data than the other structures.
The analysis of the SAXS data is difficult because the scattering length density of D2O is only about 20% higher than that of nonane which results in reduced contrast between the
D2O-rich part and nonane-rich parts of the droplet. However, we are confident that the
‘real’ droplet structure is closest to the lens-on-sphere model amongst all the structures discussed in this paper. We would need to perform small angle neutron scattering
(SANS) experiments to predict the structure with a greater accuracy because D2O and
139 nonane have much different scattering length density with respect to neutrons as opposed to X-rays.
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142 Chapter 6 Conclusions and Future Work
This chapter summarizes the conclusions of my research. It discusses qualitatively the advantages of using a supersonic separator to remove water and higher alkanes. This chapter also suggests future work required to advance this research forward.
143 We studied the condensation of nonane, D2O and their mixtures in N2 carrier gas. The condensation occurred by expanding supersaturated vapors in high speed supersonic flows with Mach number less than 3 and the residence time of the droplets in the supersonic nozzle was less than 200 μs. The preliminary data analysis used PTM which gave initial estimates of temperature, density, velocity and amount of material condensed.
We then characterized the droplet sizes and their number densities by performing position resolved SAXS measurements. The analysis of FTIR experiments yielded the composition of the aerosol which is an important parameter when vapors of D2O and nonane condense together. We collected data every 2-3 mm which corresponded to 6-8
μs due to the high speed of the condensing flow.
The first project focused on measuring the nucleation rates which are discussed in
Section 4.2.1. Nucleation rates were measured for the vapor – liquid phase transition of nonane and D2O from their supersaturated vapors at temperatures of 217 K > T >196 K.
The new nonane data line up well with the existing experimental data measured by Ghosh and co-workers (Ghosh et al. 2010) and those of D2O are about 50% higher than those measured by Kim and co-workers (Kim et al. 2004). Kim and co-workers used a similar experimental set-up and used small angle neutron scattering (SANS) at the nozzle exit to measure the number density. Their measurements were not corrected for coagulation.
From position resolved SAXS measurements, we know that D2O nanodroplets coagulate reducing N vaues to about 65% of their maximum value. This largely explains the discrepancy between the nucleation rate reported in this work and that reported by Kim et al.
144 We also studied the subsequent growth of the droplets of pure nonane and pure D2O in
Chapter 3. Position resolved size measurements yielded the experimental growth rate d r /dt. We also predicted the growth rate of droplets from the ambient conditions using a Hertz-Knudsen model. Droplet temperature can be an important parameter in predicting the growth rates. We calculated the theoretical droplet temperatures, Td,exact by solving an implicit equation based on equating the theoretical heat flux and mass flux and compare them to the experimental droplet temperatures, Td,exp using a technique which we developed by measuring the heat flux per droplet. Although for nonane, Td,exact were about 35% lower than Td,exp, droplet growth rates for nonane are insensitive to droplet temperatures under our experimental conditions. This is because for nonane, the evaporation rate of monomers from warm droplets is still negligible as compared to the condensation rate of the monomers on the droplet. Even if we assume that the droplet temperatures are same as the ambient temperatures, the predicted growth rates were higher by less than 5% . We also observed that these predicted rates match with the experimental growth rates.
In contrast to nonane, the growth of pure D2O nanodroplets was sensitive to Td because the evaporation rate of monomers was comparable to the condensation rate. It was, therefore, important that we get an accurate estimate of the droplet temperatures. For
D2O, the values of Td,exact were about 30-35% lower than Td,exp. By researching the work of Walter Drisdell (Drisdell et al. 2008) who found that the evaporation coefficient qe for rapidly evaporating water microdroplets is 0.57, we reduced the qe of D2O nanodroplets to 0.5 0.1. We also incorporated droplet growth due to coagulation which
145 we observe experimentally when we have position resolved measurements for N from
SAXS. These accommodations in the Hertz-Knudsen model resulted in the accurate predictions of droplet temperatures and the growth rates.
These experimental results of growth of D2O and nonane nanodroplets can help improve the modeling of droplet growth in supersonic nozzles. Earlier 1-D modeling work by
Sinha and co-workers (Sinha et al. 2009) suggested that isothermal growth laws are better than non-isothermal growth laws in predicting the final droplet sizes and number densities which was rather surprising. The 1-D models does not incorporate coagulation and the changes in boundary layer from the heat released during condensation and they
Future work should include the improvement of these models by incorporating 2-D and
3-D effects that track enough particle bins.
In Chapter 4 we investigated condensation when supersaturated vapors of nonane and
D2O condense together. There were nine combinations of experimental conditions for these experiments which were taken from combining three cases of pure nonane and three cases of pure D2O. In seven of these cases, D2O dominated nucleation i.e. D2O was the first species to nucleate because we observed that nucleation occurred at the same temperature as that reached by pure D2O with the same stagnation pressure and the saturation ratio required for nonane to condense at that temperature was higher than the observed saturation ratio of nonane. The nucleation rates of D2O in the presence of nonane were less than pure D2O nucleation rates at a similar T and partial pressure pv.
Although this decrease is unusual in the context of binary nucleation, it can be explained qualitatively by the non-isothermal effects of nucleation where the presence of nonane 146 vapor instead of the nitrogen adversely affects the heat transfer from the critical cluster and results in a lower nucleation rate. In one of the binary case, nonane dominated nucleation. Although the nucleation is initiated by nonane, the saturation ratio reached by
D2O was high enough such that the predicted nucleation rates of D2O from Barbara
Hale’s scaling model were within a factor of two of the nucleation rates predicted for nonane. This explains why the number densities in the binary condensation case were higher than the number densities where pure nonane at similar pv0 is condensing.
The subsequent condensation after nucleation was dominated by nonane in all the cases which we studied and we observed that condensation of D2O was inhibited by the condensation of nonane. This inhibition in condensation of D2O occurred partly because more than 90 percent of the surface area of the droplet consisted of hydrophobic nonane molecules and partly because the hotter droplets had an evaporation rate of D2O molecules which was comparable to the condensation rate of D2O molecules. This inhibition in condensation of D2O in the presence of nonane indicated that the dehydration process in supersonic separators may not be efficient unless multicomponent growth models which are used to optimize the performance of these separators incorporate this inhibition effect.
Finally, Chapter 5 explored the droplet structure of aqueous-alkane nanodroplets. These droplets were not well-mixed which was not surprising considering the immiscibility of alkanes and water. We also concluded that the droplets did not have a D2O core-nonane shell structure which we had expected because D2O has a surface tension which is three times that of nonane. Density functional theory calculations and molecular dynamics 147 simulations were conducted by our collaborators, Gerald Wilemski and his co-workers, to understand the droplet structure. The closest droplet structure to reality which we can predict is a lens-on-sphere structure where nonane lens forms a cover over a sphere of
D2O. They have also simulated the scattering pattern for such a structure by calculating the form factor. I have incorporated polydispersity into the calculation and used the non- linear optimization program provided by Argonne National Lab and fit my experimental data to the predicted scattering pattern. This prediction matched our experimental scattering data quite well. A further Guinier analysis also complimented our earlier analysis and gave values of droplet sizes and number densities which were within 15% of each other. However, the amount of D2O condensed predicted by these structures was generally 30-40% lower than that predicted by FTIR analysis. The poor contrast between nonane and D2O with regards to the X-ray scattering length density is not enough for the models to distinguish between the different regions of the composite drop. A better method to probe the droplet structure would involve conducting small angle neutron scattering (SANS) experiments. They would be helpful because D2O has a scattering length density that is an order of magnitude higher than that of nonane and the scattering behavior of neutrons from the D2O-rich region would be significant different from the nonane-rich region.
References:
1. Drisdell, W. S., Cappa, C. D., Smith, J. D., Saykally, R. J. and Cohen, R. C.
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148 2. Ghosh, D., Bergmann, D., Scwering, R., Wölk, J., Strey, R., Tanimura, S. and Wyslouzil, B. E. Homogeneous nucleation of a homologous series of n-alkanes (CiH2i+2, i = 7 - 10) in a supersonic nozzle. Journal of Chemical Physics 132, 024307-1 to 17 (2010).
3. Kim, Y. J., Wyslouzil, B. E., Wilemski, G., Wölk, J. and Strey, R. Isothermal nucleation rates in supersonic nozzles and the properties of small water clusters. Journal of Physical Chemistry A 108, 4365-4377 (2004).
4. Sinha, S., Wyslouzil, B. E. and Wilemski, G. Modeling of H2O/D2O condensation in supersonic nozzles. Aerosol Science and Technology 43, 9-24 (2009).
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25. Takahama, S. and Russell, L. M. A molecular dynamics study of water mass accommodation on condensed phase water coated by fatty acid monolayers. Journal of Geophysical Research: Atmospheres 116, D02203 (2011).
26. Tanimura, S., Wyslouzil, B. E. and Wilemski, G. CH3CH2OD/D2O binary condensation in a supersonic Laval nozzle: Presence of small clusters inferred from a macroscopic energy balance. Journal of Chemical Physics 132, 144301- 144322 (2010).
27. Viisanen, Y. and Strey, R. (1996). Composition of critical clusters in ternary nucleation of water--n-nonane--n-butanol. Journal of Chemical Physics 105, 8293-8299 (1996).
28. Wagner, P. E. A constant-angle mie scattering method (CAMS) for investigation of particle formation processes. Journal of Colloid and Interface Science 105, 456-467 (1985).
29. P.E. Wagner and R. Strey, Observation of Steady-State Aerosol Formation by Nucleation in Vapor Mixtures during Adjustable Expansion Pulses , Aerosols in Science, Industry and Environment Proceedings of the 3rd International Aerosol Conference, Kyoto, Edited by S. Masuda et al., 201-204 (1990).
164 30. Wagner, P. E. and Strey, R. (2001). Two-Pathway Homogeneous Nucleation in Supersaturated Water−n-Nonane Vapor Mixtures. The Journal of Physical Chemistry B 105, 11656-11661 (2001).
31. J. Wedenkind, A. Hyvärinen, D. Brus and D. Reguera. Unravelling the "Pressure Effect" in Nucletion. Physcial Review Letters 101, 125703-1 to 4 (2008).
32. Wyslouzil, B., Wilemski G., Strey R., Heath C. and Dieregsweiler U. Experimental evidence for internal structure in aqueous-organic nanodroplets. Physical Chemistry Chemical Physics 8, 54-57 (2006).
References for chapter 5:
1. A. Bhabhe, H. Pathak, and B. E. Wyslouzil. Freezing of heavy Water (D2O) Nanodroplets, Journal of Physical Chemistry A 117, 5472-5482 (2013).
2. Chen, B., Siepmann, J. I. and Klein, M. L. Simulating the Nucleation of Water/Ethanol and Water/n-Nonane Mixtures: Mutual Enhancement and Two- Pathway Mechanism. Journal of the American Chemical Society 125, 3113-3118 (2003).
3. Charlson, R. J., Seinfeld, J. H., Nenes, A., Kulmala, M., Laaksonen, A. and Facchini, M. C.. Reshaping the Theory of Cloud Formation. Science 292, 2025- 2026 (2001).
4. Eliason, T. L., Gilman, J. B. and Vaida, V. Oxidation of organic films relevant to atmospheric aerosols. Atmospheric Environment 38, 1367-1378 (2004) .
5. Ellison, G. B., Tuck, A. F. and Vaida, V. Atmospheric processing of organic aerosols. Journal of Geophysical Research: Atmospheres 104, 11633-11641 (1999).
165 6. Facchini, M. C., Mircea, M., Fuzzi, S. and Charlson, R. J. Cloud albedo enhancement by surface-active organic solutes in growing droplets. Nature 401, 257-259 (1999).
7. Fütterer, T., Vliegenthart, G. A. and Lang, P. R. Particle Scattering Factor of Janus Micelles. Macromolecules 37, 8407-8413 (2004).
8. Hrahsheh, F. and Wilemski, G. Fluctuating structure of aqueous organic nanodroplets. AIP Conference Proceedings 1527, 63-66 (2013).
9. S. Kline, Journal of Applied Crystallography 39, 895-900 (2006).
10. H. Laksmono, S. Tanimura, H. Allen, G. Wilemski, M. Zahniser, J. Shorter, D. Nelson, B. McManus, and B. E. Wyslouzil, Physical Chemistry Chemical Physics 13, 5855-5871 (2011).
11. Li, J.-S. and Wilemski, G. A structural phase diagram for model aqueous organic nanodroplets. Physical Chemistry Chemical Physics 8, 1266-1270 (2006).
12. A. Manka, H. Pathak, S. Tanimura, J. Wölk, R. Strey, and B.E. Wyslouzil, Physical Chemistry Chemical Physics 14, 4505-4516 (2012).
13. M. E. McKenzie and B. Chen, Journal of Physical Chemistry B 110, 3511-3516 (2006).
14. V. Modak, H. Pathak, M. Thayer, S. J. Singer, and B.E. Wyslouzil, Physical Chemistry Chemical Physics 15, 6783-6795 (2013).
15. R. B. Nellas, B. Chen, and J. I. Siepmann, Physical Chemistry Chemical Physics 9, 2779-2781 (2007).
16. H. Pathak, B. E. Wyslouzil, A. Obeidat, and G. Wilemski, AIP Conference Proceedings 1527, 472-475 (2013).
17. H. Pathak, J. Wolk, R. Strey, and B. E. Wyslouzil, AIP Conference Proceedings 1527, 51-54 (2013).
166 18. H. Pathak, K. Mullick, S. Tanimura, and B. E. Wyslouzil, Aerosol Science and Technology 47, 1310-1324 (2013).
19. H. Pathak, J. Wolk, R. Strey, and B. .E. Wyslouzil, Co-condensation of Nonane
and D2O in a supersonic nozzle, manuscript submitted to Journal of Chemical Physics.
20. P. Peters, G. Pieterse and M. E. H. van Dongen. Multi-component droplet growth. I. Experiments with supersaturated vapor and water vapor in methane. Physics of Fluids 16, 2567-2574 (2004).
21. P. Peters, G. Pieterse and M. E. H. van Dongen. Multi-component droplet growth. II. A theoretical model. Physics of Fluids 16, 2575-2586 (2004).
22. Rudich, Y. Laboratory perspectives on the chemical transformations of organic matter in atmospheric particles. Chemical Reviews 103, 5097-5124 (2003).
23. Tervahattu, H., Juhanoja, J., Vaida, V., Tuck, A. F., Niemi, J. V., Kupiainen, K., Kulmala, M. and Vehkamäki, H. Fatty acids on continental sulfate aerosol particles. Journal of Geophysical Research: Atmospheres 110, D06207 (2005).
24. Tervahattu, H., Hartonen, K., Kerminen, V.-M., Kupiainen, K., Aarnio, P., Koskentalo, T., Tuck, A. F. and Vaida, V. New evidence of an organic layer on marine aerosols. Journal of Geophysical Research: Atmospheres 107, AAC 1-1- AAC 1-8 (2002).
25. Wilemski, G., Obeidat, A. and Hrahsheh, F. Form factors for Russian doll droplet models. AIP Conference Proceedings 1527, 144-147 (2013).
26. B. E. Wyslouzil, G. Wilemski, R. Strey, C. Heath, and U. Dieregsweiler, Physical Chemistry Chemical Physics 8, 54-57 (2006).
167 27. B.E. Wyslouzil, H. Pathak, F. Hrahsheh, and G. Wilemski. Non-spherical structure of aqueous-organic nanodroplets, Bulletin of the American Physical Society 57, 2012.
References for chapter 6:
1. Drisdell, W. S., Cappa, C. D., Smith, J. D., Saykally, R. J. and Cohen, R. C.
Determination of the evaporation coefficient of D2O. Atmospheric Chemistry and Physics 8, 6699-6706 (2008).
2. Ghosh, D., Bergmann, D., Scwering, R., Wölk, J., Strey, R., Tanimura, S. and Wyslouzil, B. E. Homogeneous nucleation of a homologous series of n-alkanes (CiH2i+2, i = 7 - 10) in a supersonic nozzle. Journal of Chemical Physics 132, 024307-1 to 17 (2010).
3. Kim, Y. J., Wyslouzil, B. E., Wilemski, G., Wölk, J. and Strey, R. Isothermal nucleation rates in supersonic nozzles and the properties of small water clusters. Journal of Physical Chemistry A 108, 4365-4377 (2004).
4. Sinha, S., Wyslouzil, B. E. and Wilemski, G. Modeling of H2O/D2O condensation in supersonic nozzles. Aerosol Science and Technology 43, 9-24 (2009).
References for Appendix B:
1. Hill, P. G.; MacMillan, R. D. C.; Lee, V. A Fundamental Equation of State for Heavy Water. Journal of Physical and Chemical Reference Data 11, 1-14 (1982).
2. Tanimura, S.; Wyslouzil, B. E.; Wilemski, G. CH3CH2OD/D2O Binary Condensation in a Supersonic Laval Nozzle: Presence of Small Clusters Inferred from a Macroscopic Energy Balance. Journal of Chemical Physics 132, 1-22 (2010).
168 3. Abraham, S. F.; Lester, H. High-Speed Machine Computation of Ideal Gas Thermodynamic Functions. I. Isotopic Water Molecules. Journal of Chemical Physics 22, 2051-2058 (1954).
4. Hill, P. G.; Chris, R. D. Saturation States of Heavy Water. Journal of Physical and Chemical Reference Data 9, 735-750 (1980).
5. Wölk, J.; Strey, R. Homogeneous Nucleation of H2O and D2O in Comparison: The Isotope Effect. Journal of Physical Chemistry B 105, 11683-11701 (2001).
6. Frank, J. M.; Fred, K. L. Isothermal Compressibility of Deuterium Oxide at Various Temperatures. Journal of Chemical Physics 54, 946-949 (1971).
7. Joseph, J. J. The Surface Tension of Pure Liquid Compounds. Journal of Physical and Chemical Reference Data 1, 841-1010 (1972).
8. D.R. Lide. Handbook of Chemistry and Physics, 84th ed(CRC, Boston, 2003).
9. K. Růžička and V. Majer. Simultaneous Treatment of Vapor Pressures and Related
Thermal Data between the Triple and Normal Boiling Temperatures of n-Alkanes C5-
C20. Journal of Physical and Chemical Reference Data 2, 1-39 (1994).
10. I. Cibulka. Saturated liquid densities of 1-alkanols from C1 to C10 and n-alkanes from
C5 to C16. A critical evaluation of experimental data. Fluid Phase Equlibria 89, 1-18 (1993).
11. M.M. Rudek; J. Fisk; V.M. Chakarov and J. Katz. Condensation of a supersaturated vapor. XII. The homogeneous nucleation of the n-alkanes. Journal of Chemical Physics 105, 4707-4713 (1996).
12. Coplen, T. B. Atomic Weights of the Elements 1995 (Reprinted from International Union of Pure and Applied Chemistry from Pure Appl. Chem. vol 68, pg 2339, 1996). Journal of Physical and Chemical Reference Data 26, 1239-1253 (1997)
169 13. Span, R.; Lemmon, E. W.; Jacobsen, R. T.; Wagner, W.; Yokozeki, A. A Reference Equation of State for the Thermodynamic Properties of Nitrogen for Temperatures from 63.151 To 1000 K and Pressures to 2200 MPa. Journal of Physical and Chemical Reference Data 29, 1361-1433 (2000).
14. Benedict, M. Pressure, Volume, Temperature Properties of Nitrogen at High Density. II. Results Obtained by a Piston Displacement Method. Journal of American Chemical Society 59, 2233-2242 (1937).
15. Mage, D. T.; Jones, J. M. L.; Katz, D. L.; Roebuck, J. R. Experimental Enthalpies For Nitrogen. Chemical Engineering Progress Symposium Series No. 44, 59, 61 (1963).
16. NIST (2002). NIST Standard Reference Database REFPROP 23, in Reference Fluid Thermodynamic and Transport Properties.
References for Appendix E:
1. Wilemski, G., Obeidat, A. and Hrahsheh, F. Form factors for Russian doll droplet models. AIP Conference Proceedings 1527, 144-147 (2013).
170 Appendix A: Appendix to Chapter 2: Fortran Code to Analyze Pressure Trace Measurements
171 A.1 FORTRAN code used to calculate the temperature T, density ρ, mass fraction condensed g and velocity u using p and (A/A*)dry as input
ccc this program version includes the ability to update the latent heat as a c function of temperature for h2o and d2o based on clausius- clapeyron c approximations to liquid-vapor equations. !chhfeb2001 ccc this version of the program calculates a "wet" isentrope based on the c measured dry isentrope and corrected for the differences in gamma. it c also starts the wet condensing flow integration on the desired data c point rather than on the wet isentrope to avoid any extraneous extra c shifts. ccc smoothes all of the good density data first, then integrates from an c initial value using finer integration grid (up to 5x) ccc modified to take in pressure data instead of density data ....jul97, jlc ccc 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 ccc this version has been modified for Nozzle H on train B with Velmex (PP) ccc RTD probe is calibrated and temperature calibration factor is included ccc Now nu.dat has "tempcal" and this program reads in the value and does c temperature calibration as "to(i)=to(i)+tempcal"...... jun02, PP ccc fc=g *wi/(w10+w20) was replaced by fc=g/(w10+w20) March04 Shinobu ccc Inert gas is a mixture of N2 and CH4 March04 Shinobu ccc tisd=pp0d(i)**c0*t0 was replaced by pp0d(i)**c3*t0 July05 Shinobu ccc Function fk has been corrected 3/31/2007 Shinobu, Hartawan ccc See Vol.6, p9 and Vol. 9, p68 ccc tempcal is not used from cal07. 10/17/2007 Shinobu ccc Gas constant was set to 8.3145. 10/18/2007 Shinobu ccc Changing for Nonane-D2O, new Properties for Nonane 17/06/2009
172 implicit real*8(a-h,o-z) real*8 fcon real*8 msq,msqw,mssq real*8 rg, pi, avog real*8 dotm,dotncal,pc10,pc20,zc10 real*8 p0, t0, tempcal real*8 xstart, xthroat real*8 tt0(1000),fc(1000),g(1000),u(1000), *rr0(1000),pp0(1000),pp0d(1000) real*8 tt0_is(1000),t_is(1000),rr0_is(1000),pp0_is(1000) real*8 t_is_s(1000),pp0_is_s(1000) real*8 aratio(1000),wg(1000),t(1000),tisd(1000) real*8 xd(1000),xw(1000),x(1000) real*8 dry(1000),dryf(1000),sdry(1000) real*8 wet(1000),swet(1000),wetf(1000) real*8 po(200),p(200),deltapo(200),deltap(200),to(200) real*8 deltadry(1000),deltadryf(1000) real*8 deltawet(1000),deltawetf(1000),dtemp(1000,20) real*8 m_1,mssq_is,m_0,m_2 real*8 mdry(1000) real*8 t_is_up(1000) character*30 dryfil,wetfil,a character*8 specie(2) character*60 progname c character*4 title(3,2) common /xval/ xs(1000)
*------nomenclature c dhc,fdhc(zc10,t(i)) latent heat of condensible vapor c pc10,pc20 condensible vapor pressure (read in 2*Torr, works in dyne/cm^2) c t(i) Temperature of inert in Kelvin c zc10 Initial molar fraction of condensible vapor1 (zc10+zc20=1) *------nomenclature
progname='nuetodd2o_irCH4MFC_DryCp2Up_cal07'
open(5,file='nu_CH4_MFC.dat',status='old') ! March04 Shinobu open(10,file='4pp.out',status='unknown') open(11,file='wilson.out',status='unknown') c open(12,file='legend1.bat',status='unknown') open(13,file='dtemp.out',status='unknown') c open(14,file='legend3.bat',status='unknown') c open(15,file='legend4.bat',status='unknown') open(7,file='upstream.out',status='unknown')
call echo
pi=3.14159d0
173 rg=8.3145d7 avog=6.022d23 c read two condensible species 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,*)tempcal !PP02 !RTD probe calibration added write(*,*)'tempcal = ',tempcal,' (not used)' c convert pressures to dyn/cm**2 pconv = 760.d0/1.01325d6 c read molecular weights of carrier (1), condensible (2,3) and CH4 (4) read(5,*)wmN,wm2,wm3,wm4 ! March04 Shinobu c read specific heats of gases read(5,*)cpN,cp2,cp3,cp4 ! March04 Shinobu c read latent heat, and specific heat of condensate read(5,*)dhc2,dhc3,cpc2,cpc3 c read starting value and the number of points in the output read(5,*) xstart2, ilast2 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. c istart >= 2, ifin < ilast, ni=1 (for useless roop, do k=1,ni) read(5,*)istart2, ifin2, 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 c read x values and all of the dry data p0, p(x), and the associated standard deviations. c correct the pressures i.e. baratron calibrations and pressure loss due to mesh. 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
dotncal=0.d0 p0dry=0.d0 c t0dry=273.15d0 +tempcal t0dry=0.0d0
174 do i=1,idend read(4,*)xd(i),po(i),deltapo(i),p(i),deltap(i),to(i),dummy2, * dummy3,dummy4,flowmain,dummy4,flowsub ! xd(i) in 0.01mm, po in 2*torr c po(i)=(po(i)*0.49967 + 2.19)-poloss !Shinobu cal04 c po(i)=po(i)/1.9888 !Shinobu cal05 c po(i)=po(i)*0.5025d0+0.457d0 !Shinobu cal07 c to(i)=to(i)*0.99147d0+0.510d0 ! RuiYang and Shinobu cal07 to(i)=to(i)*0.99375d0+0.352d0 ! Shinobu cal08 Vol.11, p.33 ccc Following equation includes the effect of poloss ccc po(i)=po(i)*0.5028d0+0.732d0 ! Shinobu cal08 Vol.11, p.54 c 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 dotncal=dotncal+(flowmain+flowsub)/idend/22.41d0 p0dry=p0dry+po(i) t0dry=t0dry+to(i) enddo close(unit=4) p0dry=p0dry/idend t0dry=t0dry/idend+273.15d0 c read the number of wet data sets and flow rate of CH4 read(5,*)ndata, dotCH4
wm0dry=(dotncal*wmN+dotCH4*wm4)/(dotncal+dotCH4) ! March04 Shinobu w40dry=dotCH4*wm4/(dotncal*wmN+dotCH4*wm4) y30dry=dotCH4/(dotncal+dotCH4) cp0dry=( dotncal*wmN*cpN+dotCH4*wm4*fcp4(t0dry,p0dry,y30dry) ) * / (dotncal*wmN+dotCH4*wm4) c write(*,*)'wm0dry,w40dry,y30dry,cp0dry=', c & wm0dry,w40dry,y30dry,cp0dry write(11,1302)dryfil
do kd = 1,ndata
read(5,*)ntype,entry1,entry2,entry3 ! March04 Shinobu if(ntype.eq.0)then !pressure input (torr) pc10=entry1 pc20=entry2 tCH4=entry3 ! March04 Shinobu else if(ntype.eq.1)then !massflow and weight fraction input
175 dotm=entry1 wfc10=entry2 wfc20=1.0d0-wfc10 c write(7,*)wfc10 tCH4=entry3 ! March04 Shinobu else write(*,*)'need to specify pressure (0)' write(*,*)'or mass flow with first weight fraction input(1)' stop end if 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. tN=0.d0 t0set=0.d0 do i=1,idenw read(4,*)xw(i),po(i),deltapo(i),p(i),deltap(i),to(i),dummy1, * dummy2,dummy3,flowmain,dummy4,flowsub !po in 2*torr c po(i)=po(i)/1.9888 !Shinobu cal05 c po(i)=po(i)*0.5025+0.457 !Shinobu cal07 ccc Following equation includes the effect of poloss ccc po(i)=po(i)*0.5028d0+0.732d0 ! Shinobu cal08 Vol.11, p.54 c t0set=t0set+to(i)/idenw c to(i)=to(i)+tempcal !ppaci02! RTD probe calibration c to(i)=to(i)*0.99147d0+0.510d0 ! RuiYang and Shinobu cal07 to(i)=to(i)*0.99375d0+0.352d0 ! Shinobu cal08 Vol.11, p.33 wet(i)=p(i)/po(i) deltawet(i)=wet(i) * *((deltap(i)/p(i))**2.0+(deltapo(i)/po(i))**2.0)**0.5 c write(*,*) xw(i), wet(i), deltawet(i) !debug tN=tN+(flowmain+flowsub)/idenw/22.41d0 enddo close(unit=4) c figure out the average stagnant pressure and temperature
p0=0.0
176 t0=0.0 do i=1,idenw p0=p0+po(i)
t0=t0+to(i) enddo p0=p0/idenw t0=t0/idenw+273.15 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'
allflux=tN+tCH4+dotm*wfc10/wm2+dotm*wfc20/wm3 wm1=(tN*wmN+tCH4*wm4)/(tN+tCH4) w40=tCH4*wm4/(tN*wmN+tCH4*wm4+dotm) y30=tCH4/allflux c !chh99 c now figure out pcondensible from calibration and average properties if(ntype.eq.1)then !now calculate pcondensible pc10=p0*dotm*wfc10/wm2/allflux ! June05 Shinobu write(*,*)'pc10= ',pc10,' torr' pc20=p0*dotm*wfc20/wm3/allflux write(*,*)'pc20= ',pc20,' torr' endif c convert pressures to dyn/cm**2 p0=p0/pconv pct0=pc10+pc20 pc10=pc10/pconv pc20=pc20/pconv if((pc10+pc20).lt.1.d-18) then zc10=0.d0 else zc10=pc10/(pc10+pc20) !chh22.02.01 endif y10=pc10/p0 y20=pc20/p0 c calculate stagnation gas mass density and condensible monomer mass c density (g/cm**3) c w2,w3 are mass fraction of condensible vapor in gas wmav=(wm1*(p0-pc10-pc20)+wm2*pc10+wm3*pc20)/p0
177 w20=wm2*pc10/p0/wmav w30=wm3*pc20/p0/wmav wi=1.d0-w20-w30 wN0=wi-w40 c gw17-2-00 assuming vapor condenses at constant composition let's define c a fictitious mean condensible vapor molecular weight wmc if((pc10+pc20).lt.1.d-18) then wmc=0.d0 else wmc=(wm2*pc10+wm3*pc20)/(pc10+pc20) endif c also let's save the inital average molecular weight wmav0=wmav
cp0= wN0*cpN+w40*fcp4(t0,p0,y30) & +w20*fcp2(t0,p0,y10)+w30*fcp3(t0,p0,y20)
gamma=cp0dry/(cp0dry-rg*1.d-7/wm0dry) !n2 gamma gamma0=cp0/(cp0-rg*1.d-7/wmav) !initial mixture gamma rhog0=p0/rg/t0*wmav write(*,*)'wmav',' w20',' w30',' wi',' cp0',' gamma0' !chh061098 write(*,*)wmav, w20,w30,wi,cp0,gamma0 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) c2 = (gamma0+1.d0)/2.d0 c0 = (gamma0-1.d0)/gamma0 c3 = (gamma-1.d0)/gamma 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 tstt0=pstp0**c3
178 *********** Values at throat under Dry condition, Shinobu *************
stepm=250.0 nstepm=250 dm_1=1.d0/stepm m_1=0.d0+dm_1
t_0=t0dry p_0=p0dry
t_1=t_0 p_1=p_0 g_1=gamma cp_1=cp0dry
do i=2,nstepm
fk_1=( dgdt(t_1,p_1,0.d0,0.d0,y30dry,0.d0,0.d0,w40dry,wm0dry) & *t_1 + p_1*g_1/(g_1-1.d0)* & dgdp(t_1,p_1,0.d0,0.d0,y30dry,0.d0,0.d0,w40dry,wm0dry) )/ & ( 2.d0+(g_1-1.d0)*m_1*m_1 ) dg_1=-2.d0*fk_1*m_1*(g_1-1.d0)*2.d0*dm_1/ & (1.d0+fk_1*m_1*m_1*(g_1-1.d0)/g_1) dt_1=-t_1/(2.d0+(g_1-1.d0)*m_1*m_1)* & ( 2.d0*(g_1-1.d0)*m_1*2.d0*dm_1 + & m_1*m_1*(g_1-1.d0)/g_1*dg_1 ) dp_1=p_1/t_1*g_1/(g_1-1.d0)*dt_1 dr_1=r_1/t_1/(g_1-1.d0)*dt_1
t_2=t_0+dt_1 p_2=p_0+dp_1 r_2=r_0+dr_1 cp_2=(1.d0-w40dry)*cpN+w40dry*fcp4(t_2,p_2,y30dry) g_2=cp_2/(cp_2-rg*1.d-7/wm0dry)
t_0=t_1 p_0=p_1 t_1=t_2 p_1=p_2 cp_1=cp_2 g_1=g_2 m_1=m_1+dm_1 c write(*,*)'m_1,cp_1,g_1',m_1,cp_1,g_1 c write(*,*)'dg_1, dt_1, dp_1, ',dg_1,dt_1,dp_1 c write(*,*)'p_1/p0, t_1/t0',p_1/p0,t_1/t0
enddo
179 tstcpdry=t_1 pstcpdry=p_1
write(*,*) ' pstp0, tstt0 (for constant Cp) =', pstp0,tstt0 write(*,*) 'pstcpdry/p0dry, tstcpdry/t0dry = ', & pstcpdry/p0dry, tstcpdry/t0dry ******************************************************************** ***
pstp0dry=pstcpdry/p0dry do i=1,idend c write(*,*) i, xd(i), dry(i) !debug if((dry(i).gt. pstp0dry).and.(dry(i+1).le. pstp0dry))then c write(*,*) 'true' !debug xthroat=( pstp0dry-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 do i=1,idend x(i)=(xd(i)-xthroat)/1000.0 ! in units of cm !Shinobu for Velmex on Train B enddo c now do linear interpolation to get fixed x intervals ************************************************ ixstart=1 xstart= int(x(1)*10.d0)/10.d0 ilast=ilast2+int( (xstart2-xstart)/0.1+0.1 ) write(*,*) 'xstart= ',xstart ************************************************ 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
180 enddo
write(*,*) 'can not interpolate for point', j
5 continue enddo c we now have an array dryf(j) at fixed xs(j) intervals. now put 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) 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
*********** Values at throat under Wet condition, Shinobu *************
stepm=250.0 nstepm=250 dm_1=1.d0/stepm m_1=0.d0+dm_1
t_0=t0 p_0=p0 r_0=rhog0
181 t_1=t_0 p_1=p_0 r_1=r_0 g_1=gamma0 cp_1=cp0
do i=2,nstepm
fk_1=( dgdt(t_1,p_1,y10,y20,y30,w20,w30,w40,wmav0) & *t_1 + p_1*g_1/(g_1-1.d0)* & dgdp(t_1,p_1,y10,y20,y30,w20,w30,w40,wmav0) )/ & ( 2.d0+(g_1-1.d0)*m_1*m_1 ) dg_1=-2.d0*fk_1*m_1*(g_1-1.d0)*2.d0*dm_1/ & (1.d0+fk_1*m_1*m_1*(g_1-1.d0)/g_1) dt_1=-t_1/(2.d0+(g_1-1.d0)*m_1*m_1)* & ( 2.d0*(g_1-1.d0)*m_1*2.d0*dm_1 + & m_1*m_1*(g_1-1.d0)/g_1*dg_1 ) dp_1=p_1/t_1*g_1/(g_1-1.d0)*dt_1 dr_1=r_1/t_1/(g_1-1.d0)*dt_1
t_2=t_0+dt_1 p_2=p_0+dp_1 r_2=r_0+dr_1 cp_2= wN0*cpN+w40*fcp4(t_2,p_2,y30) & +w20*fcp2(t_2,p_2,y10)+w30*fcp3(t_2,p_2,y20) g_2=cp_2/(cp_2-rg*1.d-7/wmav)
t_0=t_1 p_0=p_1 r_0=r_1 t_1=t_2 p_1=p_2 r_1=r_2 cp_1=cp_2 g_1=g_2 m_1=m_1+dm_1 c write(*,*)'m_1,cp_1,g_1',m_1,cp_1,g_1 c write(*,*)'dg_1, dt_1, dp_1, ',dg_1,dt_1,dp_1 c write(*,*)'p_1/p0, t_1/t0',p_1/p0,t_1/t0
enddo
gammam=g_1 tstarcp=t_1 pstarcp=p_1 rstarcp=r_1 ustarcp=dsqrt(gammam*rg*tstarcp/wmav)
write(*,*)'p*/p0, t*/t0, r*/r0 (constant Cp) ', & pstp0w,1.d0/c2,c2**erho
182 write(*,*)'p_1/p0, t_1/t0, r_1/r0 ',p_1/p0,t_1/t0,r_1/rhog0 write(*,*) 'gamma0, gammam',gamma0,gammam ******************************************************************** ***
do i=1,idenw c write(*,*) i, xw(i), wet(i) !debug if((wet(i).gt.pstarcp/p0).and.(wet(i+1).le.pstarcp/p0))then c write(*,*) 'true' !debug xthroat=xw(i)+(pstarcp/p0-wet(i))/ & (wet(i+1)-wet(i))*(xw(i+1)-xw(i)) go to 5002 endif enddo 5002 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. !chh110698 ixstart=0 !chh110698 do i=1,idenw x(i)=(xw(i)-xthroat)/1000.0 ! in units of cm if(x(i).le.xstart)ixstart=i !chh110698 enddo c write(*,*) 'ixstart= ',ixstart !chh110698 write(*,*) 'throat shifted' !debug c now do linear interpolation to get fixed x intervals lasti=ixstart !chh110698 do j=1,ilast c xs values have already been assigned in dry data analysis c write(*,*) xs(j) !debug do i=lasti,idenw !chh110698 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 !chh110698 goto 6 endif enddo
write(*,*) 'can not interpolate for point', j 6 continue enddo
183 c we now have an array wetf(j) at fixed xs(j) intervals. now put 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 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'
********************************************** ifin=ifin2+int( (xstart2-xstart)/0.1+0.1 ) istart0=1 **********************************************
npts=ifin-istart0+1 nnpts=(npts-1)*nint+1 jinit=nnpts+2*nint+istart0-1 do i=ifin+1,istart0,-1 delx=xs(i)-xs(i-1) delprd = sdry(i)-sdry(i-1) c delprwi = sweti(i)-sweti(i-1) delprw = swet(i)-swet(i-1) jinit=jinit-nint jp=0
184 do j=jinit,jinit-nint+1,-1 fint=1.d0*dfloat(jp)/(1.d0*nint) xs(j)=xs(i)-delx*fint if(dabs(xs(j)).LT.1.d-4) ithroat=j pp0d(j) = sdry(i)-delprd*fint c pp0i(j) = sweti(i)-delprwi*fint pp0(j) = swet(i)-delprw *fint jp=jp+1 enddo enddo ifin1=istart0+nnpts-1
***************************************************************** istart= istart0+ & int( (xstart2-xstart)/0.1+istart2-istart0+0.1 )*nint write(*,*)'ithroat =',ithroat *****************************************************************
*** Pressure and temperature upstream of the integration region **** t_0=tstarcp p_0=pstarcp
cp_0= wN0*cpN+w40*fcp4(t_0,p_0,y30) & +w20*fcp2(t_0,p_0,y10)+w30*fcp3(t_0,p_0,y20) g_0=cp_0/(cp_0-rg*1.d-7/wmav)
dp=( pp0(ithroat+1)-pp0(ithroat-1) )*p0 dt=t_0/p_0*(g_0-1.d0)/g_0*dp
t_is_up(ithroat)=tstarcp t_is_up(ithroat+1)=tstarcp+dt/2.d0 t_is_up(ithroat-1)=tstarcp-dt/2.d0
t_1=t_is_up(ithroat+1) p_1=pp0(ithroat+1)*p0
do i=ithroat+2,istart cp_1= wN0*cpN+w40*fcp4(t_1,p_1,y30) & +w20*fcp2(t_1,p_1,y10)+w30*fcp3(t_1,p_1,y20) g_1=cp_1/(cp_1-rg*1.d-7/wmav)
dp=( pp0(i)-pp0(i-2) )*p0 dt=t_1/p_1*(g_1-1.d0)/g_1*dp
t_is_up(i)=t_is_up(i-2)+dt
t_1=t_is_up(i) p_1=pp0(i)*p0 enddo
t_1=t_is_up(ithroat-1)
185 p_1=pp0(ithroat-1)*p0
do i=ithroat-2,istart0,-1 cp_1= wN0*cpN+w40*fcp4(t_1,p_1,y30) & +w20*fcp2(t_1,p_1,y10)+w30*fcp3(t_1,p_1,y20) g_1=cp_1/(cp_1-rg*1.d-7/wmav)
dp=( pp0(i)-pp0(i+2) )*p0 dt=t_1/p_1*(g_1-1.d0)/g_1*dp
t_is_up(i)=t_is_up(i+2)+dt
t_1=t_is_up(i) p_1=pp0(i)*p0 enddo
write(7,700) 700 format(' x(cm) pp0 p(Torr) Tisw(K)')
do i=istart0,istart write(7,710) xs(i),pp0(i),pp0(i)*p0*pconv,t_is_up(i) 710 format(f7.2,f8.4,2f10.2) enddo close(unit=7) ******************************************************************** ***
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
***********Values at the start point of integration for Dry, Shinobu **********
stepp=100.0 nstepp=100 dp_1=(pp0d(istart-1)*p0dry-pstcpdry)/stepp t_0=tstcpdry p_0=pstcpdry cp_0=(1.d0-w40dry)*cpN+w40dry*fcp4(t_0,p_0,y30dry) g_0=cp_0/(cp_0-rg*1.d-7/wm0dry) m_0=1.d0 fk_0=( dgdt(t_0,p_0,0.d0,0.d0,y30dry,0.d0,0.d0,w40dry,wm0dry) & *t_0 + p_0*g_0/(g_0-1.d0)* & dgdp(t_0,p_0,0.d0,0.d0,y30dry,0.d0,0.d0,w40dry,wm0dry) )/
186 & ( 2.d0+(g_0-1.d0)*m_0*m_0 ) a_0=1.d0
dt_0=t_0/p_0*(g_0-1.d0)/g_0*dp_1 dm_0= -(2.d0+(g_0-1.d0)*m_0**2) & *( g_0+fk_0*(m_0**2)*(g_0-1.d0) ) & /(2.d0*(g_0**2)*m_0)/p_0*dp_1 da_0= -a_0*(m_0**2-1.d0)/(g_0*m_0**2)/p_0*dp_1
t_1=t_0+dt_0 m_1=m_0+dm_0 p_1=p_0+dp_1 a_1=a_0+da_0 cp_1=(1.d0-w40dry)*cpN+w40dry*fcp4(t_1,p_1,y30dry) g_1=cp_1/(cp_1-rg*1.d-7/wm0dry) fk_1=( dgdt(t_1,p_1,0.d0,0.d0,y30dry,0.d0,0.d0,w40dry,wm0dry) & *t_1 + p_1*g_1/(g_1-1.d0)* & dgdp(t_1,p_1,0.d0,0.d0,y30dry,0.d0,0.d0,w40dry,wm0dry) )/ & ( 2.d0+(g_1-1.d0)*m_1*m_1 )
do i=2, nstepp dt_1=t_1/p_1*(g_1-1.d0)/g_1*2.d0*dp_1 dm_1= -(2.d0+(g_1-1.d0)*m_1**2) & *( g_1+fk_1*(m_1**2)*(g_1-1.d0) ) & /(2.d0*(g_1**2)*m_1)/p_1*2.d0*dp_1 da_1= -a_1*(m_1**2-1.d0)/(g_1*m_1**2)/p_1*2.d0*dp_1
t_2=t_0+dt_1 m_2=m_0+dm_1 a_2=a_0+da_1 p_2=p_0+2.d0*dp_1
cp_2=(1.d0-w40dry)*cpN+w40dry*fcp4(t_2,p_2,y30dry) g_2=cp_2/(cp_2-rg*1.d-7/wm0dry) fk_2=( dgdt(t_2,p_2,0.d0,0.d0,y30dry,0.d0,0.d0,w40dry,wm0dry) & *t_2 + p_2*g_2/(g_2-1.d0)* & dgdp(t_2,p_2,0.d0,0.d0,y30dry,0.d0,0.d0,w40dry,wm0dry) )/ & ( 2.d0+(g_2-1.d0)*m_2*m_2 )
t_0=t_1 m_0=m_1 a_0=a_1 p_0=p_1 t_1=t_2 m_1=m_2 a_1=a_2 g_1=g_2 fk_1=fk_2
187 p_1=p_2 c write(*,*)'dt_1,dm_1,da_1 =',dt_1,dm_1,da_1
enddo
tisd(istart-1)=t_1 aratio(istart-1)=a_1 mdry(istart-1)=m_1
dp_2=dp_1+(pp0d(istart)-pp0d(istart-1))*p0dry dt_1=t_1/p_1*(g_1-1.d0)/g_1*dp_2 dm_1= -(2.d0+(g_1-1.d0)*m_1**2) & *( g_1+fk_1*(m_1**2)*(g_1-1.d0) ) & /(2.d0*(g_1**2)*m_1)/p_1*dp_2 da_1= -a_1*(m_1**2-1.d0)/(g_1*m_1**2)/p_1*dp_2
t_2=t_0+dt_1 m_2=m_0+dm_1 a_2=a_0+da_1 p_2=p_0+dp_2
tisd(istart)=t_2 mdry(istart)=m_2 aratio(istart)=a_2 ******************************************************************** ************* 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. c msqw = c10*((1.d0/pp0(istart-1))**c0-1.d0) c tt0(istart-1)=1.d0/(1.d0+msqw/c10) c rr0(istart-1)=(1.d0+msqw/c10)**erho c msqw = c10*((1.d0/pp0( istart))**c0-1.d0) c tt0(istart)=1.d0/(1.d0+msqw/c10) c rr0(istart)=(1.d0+msqw/c10)**erho
***********Values at the start point of integration for Wet, Shinobu **********
stepp=100.0 nstepp=100 dp_1=(pp0(istart-1)*p0-pstarcp)/stepp
t_0=tstarcp p_0=pstarcp r_0=rstarcp
188 g_0=gammam
dt_0=t_0/p_0*(g_0-1.d0)/g_0*dp_1 dr_0=r_0/p_0/g_0*dp_1 t_1=t_0+dt_0 r_1=r_0+dr_0 p_1=p_0+dp_1 cp_1= wN0*cpN+w40*fcp4(t_1,p_1,y30) & +w20*fcp2(t_1,p_1,y10)+w30*fcp3(t_1,p_1,y20) g_1=cp_1/(cp_1-rg*1.d-7/wmav)
do i=2,nstepp
dt_1=t_1/p_1*(g_1-1.d0)/g_1*2.d0*dp_1 dr_1=r_1/p_1/g_1*2.d0*dp_1 t_2=t_0+dt_1 r_2=r_0+dr_1 p_2=p_0+2.d0*dp_1 cp_2= wN0*cpN+w40*fcp4(t_2,p_2,y30) & +w20*fcp2(t_2,p_2,y10)+w30*fcp3(t_2,p_2,y20) g_2=cp_2/(cp_2-rg*1.d-7/wmav)
t_0=t_1 r_0=r_1 p_0=p_1 t_1=t_2 r_1=r_2 cp_1=cp_2 g_1=g_2 p_1=p_2
enddo
tt0_is(istart-1)=t_1/t0 rr0_is(istart-1)=r_1/rhog0 pp0_is(istart-1)=p_1/p0
msqw = c10*((1.d0/pp0(istart-1))**c0-1.d0) write(*,*)'pp0(istart-1), tt0, rr0', & pp0(istart-1),1.d0/(1.d0+msqw/c10),(1.d0+msqw/c10)**erho write(*,*)'p_1/p0,t_1/t0,r_1/rhog0', & pp0_is(istart-1),tt0_is(istart-1),rr0_is(istart-1)
dp_2=dp_1+(pp0(istart)-pp0(istart-1))*p0 dt_1=t_1/p_1*(g_1-1.d0)/g_1*dp_2 dr_1=r_1/p_1/g_1*dp_2 t_2=t_0+dt_1 r_2=r_0+dr_1
tt0_is(istart)=t_2/t0 rr0_is(istart)=r_2/rhog0
189 pp0_is(istart)=(p_0+dp_2)/p0
msqw = c10*((1.d0/pp0( istart))**c0-1.d0) write(*,*)'pp0(istart), tt0, rr0 ', & pp0(istart),1.d0/(1.d0+msqw/c10),(1.d0+msqw/c10)**erho write(*,*)'p_2/p0,t_2/t0,r_2/rhog0', & pp0_is(istart),tt0_is(istart),rr0_is(istart)
tt0(istart-1)=tt0_is(istart-1) rr0(istart-1)=rr0_is(istart-1) tt0(istart)=tt0_is(istart) rr0(istart)=rr0_is(istart) ******************************************************************** ***
g(istart)=0.d0 g(istart-1)=0.d0 fc(istart)=0.0d0 !fraction condensed
write(10,1024) progname 1024 format('Program: ',a60)
write(10,1011)p0*pconv,devp0,t0-273.15, t0set, rhog0*1.0d3 !4pp plots write(10,1010) dotm,specie(1),wfc10 !4pp plots 1010 format('Weight flux of condensable =',f6.2, & ' g/min , Fraction of ',a, '=',f7.3) !4pp plots 1011 format('p0= ',f6.2,'+/-',f4.2,' Torr T0=',f6.2, & ' C (set T0=',f6.2,') rho0=', e11.4,' kg/m3') !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) allflux, (dotncal+dotCH4) !4pp plots 1013 format( 'Total mole flux =', & f6.3,' mol/min (',f6.3,' in Dry )' ) !4pp plots write(10,1019) tCH4, dotCH4 !4pp plots 1019 format( ' CH4 mole flux =', & f6.3,' mol/min (',f6.3,' in Dry )' ) !4pp plots write(10,1014)wetfil,dryfil !4pp plots 1014 format('@subtitle \"',a30,'with dry trace ',a30,'\"') !4pp plots write(10,1015) !4pp plots 1015 format(' x(cm) u(m/s) T(K) p/p0 Tis p/p0_is', &' MoleFract. g g/g_inf A/A* r/r0', &' Tisd p/p0_isd') !4pp plots
190 c write(12,1016)kd,pc10*pconv,specie(1),pc20*pconv, c +specie(2) !4pp plots c 1016 format('legend string ',i2,' \"',2(f7.4,'torr ',a),'\"') !4pp plots c write(14,1017)kd, wetfil !4pp plots c 1017 format('legend string ',i2,' \"',a13,'\"') !4pp plots c write(15,1018)kd-1,p0*pconv,devp0,t0-273.15 !4pp plots c 1018 format('legend string ',i2,' \"',f6.2,'+/-',f4.2,'torr ', c & f6.2,'celsius"') !4pp plots
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
*********** Integration of the isentropic curve for Dry, Shinobu **************
dp_dry=( pp0d(i+1)-pp0d(i-1) )*p0dry/2.d0
p_dry=pp0d(i)*p0dry cp_dry=(1.d0-w40dry)*cpN + w40dry*fcp4(tisd(i),p_dry,y30dry) g_dry=cp_dry/(cp_dry-rg*1.d-7/wm0dry) fk_dry=( dgdt(tisd(i),p_dry,0.d0,0.d0,y30dry, & 0.d0,0.d0,w40dry,wm0dry) & *tisd(i) + p_dry*g_dry/(g_dry-1.d0)* & dgdp(tisd(i),p_dry,0.d0,0.d0,y30dry, & 0.d0,0.d0,w40dry,wm0dry) )/ & ( 2.d0+(g_dry-1.d0)*mdry(i)**2 )
dt_dry=tisd(i)/p_dry*(g_dry-1.d0)/g_dry*dp_dry dm_dry= -( 2.d0+(g_dry-1.d0)*mdry(i)**2 ) & *( g_dry+fk_dry*(mdry(i)**2)*(g_dry-1.d0) ) & /(2.d0*(g_dry**2)*mdry(i))/p_dry*dp_dry da_dry= -aratio(i)*(mdry(i)**2-1.d0)/(g_dry*mdry(i)**2)/ & p_dry*dp_dry
tisd(i+1)=tisd(i-1)+2.d0*dt_dry mdry(i+1)=mdry(i-1)+2.d0*dm_dry aratio(i+1)=aratio(i-1)+2.d0*da_dry
dar=dlog(aratio(i+1)/aratio(i-1))/2.d0 c write(*,*) 'Integration of the isentropic curve for Dry OK'
191 ******************************************************************** ************* ccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccc c Arearatio for constant Cp of CH4 c c msq = c1*((1.d0/pp0d(i))**c3-1.d0) c aratio(i)= dsqrt(((c1/eam2*(1.d0+msq/c1))**eam2)/msq) c msq = c1*((1.d0/pp0d(i+1))**c3-1.d0) c aratio(i+1) = dsqrt(((c1/eam2*(1.d0+msq/c1))**eam2)/msq) c dar=dlog(aratio(i+1)/aratio(i-1))/2.d0 c c cccccccccccccccccccccccccccccccccccccccccccccccccccccccccccc
*********** Integration of the isentropic curve for Wet, Shinobu **************
dp=(pp0(i+1)-pp0(i-1))/2.d0
t_is(i)=tt0_is(i)*t0
u_is=ustarcp*rstarcp/rhog0/rr0_is(i)/aratio(i) mssq_is=(u_is/ustarcp)**2 cp_is= wN0*cpN+w40*fcp4(t_is(i),pp0_is(i)*p0,y30) & +w20*fcp2(t_is(i),pp0_is(i)*p0,y10) & +w30*fcp3(t_is(i),pp0_is(i)*p0,y20) cpr_is=cp_is/cp0 hpara_is=1.d0-(gamma0-1.d0)/gamma0/cpr_is
tempA=(1.d0-t_is(i)/tstarcp/gammam/mssq_is)/rr0_is(i) tempG=t0/tstarcp/gammam/mssq_is tempJ=rr0_is(i)*tt0_is(i)/ & (hpara_is-t_is(i)/tstarcp/gammam/mssq_is)
dtt0_is=(tt0_is(i)-tempA*tempJ)*dar dpp0_is=-tempJ*dar drr0_is=-(rr0_is(i)+tempG*tempJ)*dar
tt0_is(i+1)=tt0_is(i-1)+2.d0*dtt0_is pp0_is(i+1)=pp0_is(i-1)+2.d0*dpp0_is rr0_is(i+1)=rr0_is(i-1)+2.d0*drr0_is c write(*,*) 'Integration of the isentropic curve for Wet OK' *************** Smoothing, Shinobu ************************************ t_is_s(i)=t0*( tt0_is(i-1)+2.d0*tt0_is(i)+tt0_is(i+1) )/4.d0 pp0_is_s(i)=( pp0_is(i-1)+2.d0*pp0_is(i)+pp0_is(i+1) )/4.d0 c t_is_s(i)=t0*tt0_is(i) c pp0_is_s(i)=pp0_is(i) ******************************************************************** ***
192 c write(*,*)'mssq,mssq_is',mssq,mssq_is c write(*,*)'cp,cp_is',cp,cp_is c write(*,*)'hpara,hpara_is',hpara,hpara_is c write(*,*)'gamma0,gammam',gamma0,gammam c write(*,*)'tempA, tempJ,tempG',tempA,tempJ,tempG c write(*,*) rr0_is(i) c write(*,*)'dar',dar c write(*,*)'dtt0,dpp0',dtt0_is,dpp0_is c write(*,*)'mdry(i), Mach ,fk_dry=', c & mdry(i), dsqrt(c1*((1.d0/pp0d(i))**c3- 1.d0)),fk_dry
******************************************************************** *** t(i)=tt0(i)*t0
u(i)=ustarcp*rstarcp/rhog0/rr0(i)/aratio(i) mssq=(u(i)/ustarcp)**2
fcon=dotm * (wfc10/wm2 + (1-wfc10)/wm3) if((pc10+pc20).lt.1.d-18) then y1=0.d0 y2=0.d0 else y1=(1-fc(i))*fcon/(allflux-fc(i)*fcon)*y10/(y10+y20) y2=(1-fc(i))*fcon/(allflux-fc(i)*fcon)*y20/(y10+y20) endif y3=y30*allflux/(allflux-fc(i)*fcon) c write(*,*)'(y1+y2+y3+tN/allflux*y3/y30), (wN0+w20+w30+w40)', c & y1+y2+y3+tN/allflux*y3/y30, wN0+w20+w30+w40 c gw2-17-00 update specific heat if((pc10+pc20).lt.1.d-18) then cpv=0.d0 cpc=0.d0 else cpv=( w20*fcp2(t(i),pp0(i)*p0,y1) + & w30*fcp3(t(i),pp0(i)*p0,y2) )/(w20+w30) cpc=( w20*fcpl2(t(i))+w30*fcpl3(t(i)) )/(w20+w30) endif cp= wN0*cpN+w40*fcp4(t(i),pp0(i)*p0,y3) & +(w20+w30-g(i))*cpv+g(i)*cpc cpr=cp/cp0 c gw2-17-00 update "mu/(1-g)" = wmu, and related factors if((pc10+pc20).lt.1.d-18) then wmu=wm1 wg(i)=0.d0
193 else wmu=wm1*wmc/(wi*wmc+(w20+w30-g(i))*wm1) wg(i)=wmu/(wmc) endif wmuu0=wmu/wmav0
hpara=wmuu0-(gamma0-1.d0)/gamma0/cpr dr=dp/tstarcp*t0/gammam/mssq-rr0(i)*dar dgp=(hpara-t(i)/tstarcp/mssq/gammam)/rr0(i)*dp+tt0(i)*dar if((pc10+pc20).lt.1.d-18) then dg=0.d0 else dg=dgp*cp*t0/(fdhc(wfc10,t(i))-cp*t(i)*wg(i)) ! Shinobu endif c gw2-17-00 update dtt0 dtt0=(wmuu0-t(i)/tstarcp/gammam/mssq)/rr0(i)*dp+ & tt0(i)*(dar+wg(i)*dg)
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
if((w20+w30).gt.0.0)then fc(i+1)=g(i+1)/(w20+w30) ! March04 Shinobu else fc(i+1)=0.0 end if c c write(*,*) 'Integration of the Wet trace OK' write(10,1020)xs(i),u(i)/100,t(i),pp0(i),t_is_s(i),pp0_is_s(i), ! June05 Shinobu * (1-fc(i))*fcon/(allflux-fc(i)*fcon),g(i),fc(i),aratio(i), * rr0(i),tisd(i),pp0d(i)
1020 format(f8.3,f10.2,f8.2,f8.4,f8.2,f8.4,2e13.4,f8.4,f8.4,f8.4, & f8.2,f8.4) !4pp plots 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)-t_is_s(i) and t(i)-tisd
do i = istart,ifin1-1 dtemp(i,kd) = t(i)-t_is_s(i)
194 dt1 = t(i) - t_is_s(i) dt2 = t(i+1) -t_is_s(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 = pp0_is_s(i)+(0.5-dt1)/(dt2-dt1)*(pp0_is_s(i+1)- pp0(i)) ton = t(i)+(0.5-dt1)/(dt2-dt1)*(t(i+1)-t(i)) tiswon = t_is_s(i)+(0.5-dt1)/(dt2-dt1)*(t_is_s(i+1)- t_is_s(i)) else endif enddo write(*,*)'using the t-t_is_s = 0.5 k' write(*,1300)xon,pp0on*pct0,ton, & pp0ion*pct0,tiswon 1300 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,
& pp0on*pc10*pconv,pp0on*pc20*pconv,wetfil,xon 1301 format(f8.2,f8.2,f8.4,f8.2,f8.4,f8.4,f8.4,2x,a13,f7.1) 1302 format('@\"t0 p0 pct ton pon p1on p2on', & 6x,a13,'\"') xon=0.0 pp0on=0.0 ton=0.0 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)) pp0don = pp0d(i)+(0.5-dt1)/(dt2-dt1)*(pp0d(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)) else endif enddo write(*,*)'using the criterion t-tisd = 0.5 k' write(*,1300)xon,pp0on*(pc10+pc20)*pconv,ton, & pp0don*(pc10+pc20)*pconv,tisdon
write(*,*) 'finished integration' enddo enddo c now write out the dtemp files to dtemp.out
do i = istart,ifin1 write(13,1313)xs(i),(dtemp(i,j),j=1,ndata) enddo
195 1313 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 c regardless of the spacing c c m - order of the highest polynomial used in smoothing c n - number of y points in interval over which smoothing is performed c k - point whose smoothed value is desired c k0 - first point in set of n c sval - smoothed value returned to calling program c 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)
196 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(9,2)a goto 15 99 continue rewind 5 return 1 format(a100) 2 format(1x,a100) 3 format(1h1,20x,'input file',//) end *------chh22.02.01---* * c real function fdhc(dhc) c fdhc = dhc c return * real*8 function fdhc(wfc10,tk) double precision zc10,wfc10,tk,rg *------general nomenclature-* c rg universal gas constant in units of c tk temperature of vapor condensing in kelvin c zc10 molar fraction of condensible 1 in vapor (zc10+zc20=1.0) *------condensible nomenclature-* c a2h2o - a4h2o h2o vapor pressure constants, wagner correlation c a1d2o - a6d2o d2o vapor pressure constants, c mwd2o d2o molecular weight c z d2o intermediate variable *------* double precision a1d2o,a2d2o,a3d2o,a4d2o,a5d2o,a6d2o,z,mwd2o !d2o pve constants double precision bbu,cbu,mwetod,dhcetod
rg=8.3145d0
*------*
197 *-d2o clausius-clapeyron relation applied to equilibrium vapor pressure *-d2o valid for temperature range of 275-823K *-d2o hill, mcmillan, and lee, j. phys chem ref data, vol 11, no.1, p1-14 (1982) a1d2o= -7.81583d0 a2d2o= 17.6012d0 a3d2o=-18.1747d0 a4d2o= -3.92488d0 a5d2o= 4.19174d0 a6d2o=643.89d0 mwd2o=20.03d0
z=1-tk/a6d2o d2oa=a1d2o*z+a2d2o*z**1.9+a3d2o*z**2+a4d2o*z**5.5+a5d2o*z**10. d2ob=a1d2o+1.9d0*a2d2o*z**0.9+2.d0*a3d2o*z+5.5d0*a4d2o*z**4.5 &+10.d0*a5d2o*z**9. dhcd2o=-rg*(a6d2o*d2oa+tk*d2ob)/mwd2o
*------* *-BuOH clausius-clapeyron relation applied to equilibrium vapor pressure *-BuOH valid for temperature range of 243.2-303.2 K * And T. Schmeling and R. Strey, Ber. Bunsenges. Phys. Chem., vol 87, p871-874 (1983) c bbu= 9412.61d0 c cbu= 10.54d0 c mwbuOH=74.12d0 c dhcbuOH=rg*(bbu-cbu*tk)/mwbuOH c Use corrected equation 20 of Ruzicka and Majer J physical chem ref data 23, 1994 p 1-39 c Note the T is missing from the a1 term!Original units are J/mol a0 = 2.94690d0 a1 = -2.051933d-3 a2 = 1.903683d-6 Tb = 423.932d0 mwetod=128.260d0
Term1 = rg*exp(a0 + a1*tk + a2*tk*tk)
Term2 = Tb + tk*(tk - Tb)*(a1+2.0d0*a2*tk) ccc dhcetod= Hvap von Nonane!!!
dhcetod = (Term1*Term2)/mwetod
198 *------* fdhc = (wfc10*dhcetod)+(1.d0-wfc10)*dhcd2o c write(28,*)'dhc debug: fdhc= ',fdhc,' wfc10= ',wfc10,' tk= ',tk,' K' !debug dhc return
end
*------*
***************** Functions for Cp **********************************
real*8 function fcp2(tk,p,y1) double precision tk,p,y1 double precision mw,a0,a1,a2 c** Cp of BuOH ****** p: Total static pressure ****** y1: Mole fraction of condensable 1 in vapor phase c mw=74.12 c a0=30.941d0 c a1=0.10037d0 c a2=7.322d-5 c fcp2=(a0+a1*tk+a2*tk*tk)/mw (data for EtOD) c fcp2=1.473 used a const cp for BuOh?? c [J/g*K]
*----Cp Nonane--Ruzicka & Majer J.Phys.Chem.Ref.Data.Vol.23.No.1,1994 ------***NEU machen * taken from Bures et al, Chem Eng Sci, 36, 529-537 (1981) c------Einheit J/molK------
A = 148.15036d0 B1 = 288.24904d0 B2 = 178.57491d0 C1 = 1380.8003d0 C2 = 3051.1566d0
Term1 = (C1/tk)*(C1/tk)*dexp(-C1/tk)/(1.0d0 - dexp(- C1/tk))**2
Term2 = (C2/tk)*(C2/tk)*dexp(-C2/tk)/(1.0d0 - dexp(- C2/tk))**2
cpnonane = A + B1*Term1 + B2*Term2
199 mw=128.260d0
fcp2 = cpnonane/mw c unit for the code should be J/g*K
return end
real*8 function fcp3(tk,p,y2) double precision tk,p,y2 double precision rg,mw,a0,a1,a2,a3 c** Cp of D2O: Fitted to the data in JCP 22, 2051 (1954) at T= 160- 340 K rg=8.3145 mw=20.03 a0=4.1712 a1=-2.2388d-3 a2= 8.6096d-6 a3=-5.6304d-9 fcp3=rg*(a0+a1*tk+a2*tk**2+a3*tk**3)/mw c fcp3=1.710d0 return end
real*8 function fcp4(tk,p,y3) double precision tk,p,y3 double precision rg,mw,a0,a1,a2,a3 c** Cp of CH4: Fitted to the data in J. Chem. Eng. Data 8, 547 (1963) at T= 160-340 K rg=8.3145 mw=16.04 a0=4.337 a1=-3.7677d-3 a2= 9.1107d-6 a3= 1.0178d-9 fcp4=rg*(a0+a1*tk+a2*tk**2+a3*tk**3)/mw c fcp4=2.226d0 return end
************** Temperature derivative of Cp ****************
real*8 function fdcp2dt(tk,p,y1) double precision tk,p,y1,mw,a1,a2,c1,c2 c mw=74.12 c a1=0.10037d0 c a2=7.322d-5 c fdcp2dt=(a1+2.d0*a2*tk)/mw
200 c----Cp Nonane--Ruzicka & Majer J.Phys.Chem.Ref.Data.Vol.23.No.1,1994 ------***NEU machen * taken from Bures et al, Chem Eng Sci, 36, 529-537 (1981) c------Einheit J/molK------
A = 148.15036d0 B1 = 288.24904d0 B2 = 178.57491d0 C1 = 1380.8003d0 C2 = 3051.1566d0 mwnonane=128.258d0 ccc Näherung für cp/dt
tk1 = tk+ 0.1d0 tk2 = tk - 0.01
Term1 = (C1/tk1)*(C1/tk1)*dexp(-C1/tk1)/ & (1.0d0 - dexp(-c1/tk1))**2 Term2 = (C2/tk1)*(C2/tk1)*dexp(-C2/tk1)/ & (1.0d0 - dexp(-c2/tk1))**2
cpnonane1 = (A + B1*Term1 + B2*Term2)/mwnonane
Term1 = (C1/tk2)*(C1/tk2)*dexp(-C1/tk2)/ & (1.0d0 - dexp(-c1/tk2))**2 Term2 = (C2/tk2)*(C2/tk2)*dexp(-C2/tk2)/ & (1.0d0 - dexp(-c2/tk2))**2
cpnonane2 = (A + B1*Term1 + B2*Term2)/mwnonane
fdcp2dt = (cpnonane1-cpnonane2)/(0.02d0)
return end
real*8 function fdcp3dt(tk,p,y2) double precision tk,p,y2 double precision rg,mw,a1,a2,a3 c** Cp of D2O: Fitted to the data in JCP 22, 2051 (1954) at T= 160- 340 K rg=8.3145 mw=20.03 a1=-2.2388d-3 a2= 8.6096d-6 a3=-5.6304d-9 fdcp3dt=rg*(a1+2.d0*a2*tk+3.d0*a3*tk**2)/mw c fdcp3dt=0.0 return end
201 real*8 function fdcp4dt(tk,p,y3) double precision tk,p,y3 double precision rg,mw,a1,a2,a3 c** Cp of CH4: Fitted to the data in J. Chem. Eng. Data 8, 547 (1963) at T= 160-340 K rg=8.3145 mw=16.04 a1=-3.7677d-3 a2= 9.1107d-6 a3= 1.0178d-9 fdcp4dt=rg*(a1+2.d0*a2*tk+3.d0*a3*tk**2)/mw c fdcp4dt=0.0 return end
*************** Pressure derivative of Cp *********************
real*8 function fdcp2dp(tk,p,y1) double precision tk,p,y1 fdcp2dp=0.d0 return end
real*8 function fdcp3dp(tk,p,y2) double precision tk,p,y2 fdcp3dp=0.d0 return end
real*8 function fdcp4dp(tk,p,y3) double precision tk,p,y3 fdcp4dp=0.d0 return end
***************** Functions for Cpl ********************************** c** Cpl of Nonane real*8 function fcpl2(tk) double precision tk fcpl2=2.2170d0 return end c** Cpl of D2O real*8 function fcpl3(tk) double precision tk fcpl3=4.205d0
202 return end
******** Temperature derivative of gamma of gas mixture ***
real*8 function dgdt(tk,p,y1,y2,y3,w20,w30,w40,wmav0) double precision tk,p,y1,y2,y3,w20,w30,w40,wmav0 double precision rg,cpN,cp,gamma,dcpdt
rg=8.3145 cpN=1.0397
cp= (1.d0-w20-w30-w40)*cpN+w40*fcp4(tk,p,y3) & +w20*fcp2(tk,p,y1)+w30*fcp3(tk,p,y2) gamma=cp/(cp-rg/wmav0)
dcpdt= w20*fdcp2dt(tk,p,y1)+w30*fdcp3dt(tk,p,y2) & +w40*fdcp4dt(tk,p,y3)
dgdt=gamma*(1.d0-gamma)/cp*dcpdt
return end
******** Pressure derivative of gamma of gas mixture ******
real*8 function dgdp(tk,p,y1,y2,y3,w20,w30,w40,wmav0) double precision tk,p,y1,y2,y3,w20,w30,w40,wmav0 double precision rg,cpN,cp,gamma,dcpdp
rg=8.3145 cpN=1.0397
cp= (1.d0-w20-w30-w40)*cpN+w40*fcp4(tk,p,y3) & +w20*fcp2(tk,p,y1)+w30*fcp3(tk,p,y2) gamma=cp/(cp-rg/wmav0)
dcpdp= w20*fdcp2dp(tk,p,y1)+w30*fdcp3dp(tk,p,y2) & +w40*fdcp4dp(tk,p,y3)
dgdp=gamma*(1.d0-gamma)/cp*dcpdp
return end
203 A.2 FORTRAN code used to calculate the experimental parameters using p and gfit from SAXS as input ccc this program version includes the ability to update the latent heat as a c function of temperature for h2o and d2o based on clausius- clapeyron c approximations to liquid-vapor equations. !chhfeb2001 ccc this version of the program calculates a "wet" isentrope based on the c measured dry isentrope and corrected for the differences in gamma. it c also starts the wet condensing flow integration on the desired data c point rather than on the wet isentrope to avoid any extraneous extra c shifts. ccc smoothes all of the good density data first, then integrates from an c initial value using finer integration grid (up to 5x) ccc modified to take in pressure data instead of density data ....jul97, jlc ccc 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 ccc this version has been modified for Nozzle H on train B with Velmex (PP) ccc RTD probe is calibrated and temperature calibration factor is included ccc Now nu.dat has "tempcal" and this program reads in the value and does c temperature calibration as "to(i)=to(i)+tempcal"...... jun02, PP ccc fc=g *wi/(w10+w20) was replaced by fc=g/(w10+w20) March04 Shinobu ccc Inert gas is a mixture of N2 and CH4 March04 Shinobu ccc tisd=pp0d(i)**c0*t0 was replaced by pp0d(i)**c3*t0 July05 Shinobu ccc Function fk has been corrected 3/31/2007 Shinobu, Hartawan ccc See Vol.6, p9 and Vol. 9, p68 ccc tempcal is not used from cal07. 10/17/2007 Shinobu ccc Gas constant was set to 8.3145. 10/18/2007 Shinobu ccc Changing for Nonane-D2O, new Properties for Nonane 17/06/2009
implicit real*8(a-h,o-z) real*8 fcon
204 real*8 msq,msqw,mssq real*8 rg, pi, avog real*8 dotm,dotncal,pc10,pc20,zc10 real*8 p0, t0, tempcal real*8 xstart, xthroat real*8 tt0(2000),fc(2000),g(2000),u(2000), *rr0(2000),pp0(2000),pp0d(2000) real*8 tt0_is(2000),t_is(2000),rr0_is(2000),pp0_is(2000) real*8 t_is_s(2000),pp0_is_s(2000) real*8 aratio(2000),wg(1000),t(2000),tisd(2000) real*8 xd(2000),xw(2000),x(2000) real*8 dry(2000),dryf(2000),sdry(2000) real*8 wet(2000),swet(2000),wetf(2000) real*8 po(400),p(400),deltapo(400),deltap(400),to(400) real*8 deltadry(2000),deltadryf(2000) real*8 deltawet(2000),deltawetf(2000),dtemp(2000,40) real*8 m_1,mssq_is,m_0,m_2 real*8 mdry(2000) real*8 t_is_up(2000) c c ccccccccccccccccccccccccccccccc ! Shinobu ccccccccc real*8 mssq_TDL,cp_TDL,cpr_TDL real*8 fc_TDL(2000),tt0_TDL(2000),u_TDL(2000),rr0_TDL(2000) real*8 g_TDL(2000),g_TDL2(2000),g_TDL3(2000) real*8 x_TDL(2000),t_TDL(2000),ar_TDL(2000) real*8 cpv_TDL,cpc_TDL,y1_TDL,y2_TDL,y3_TDL cccccccccccccccccccccccccccccccccccccccccccccccccccccccccc
character*30 dryfil,wetfil,a character*8 specie(2) character*60 progname c character*4 title(3,2) common /xval/ xs(2000)
*------nomenclature c dhc,fdhc(zc10,t(i)) latent heat of condensible vapor c pc10,pc20 condensible vapor pressure (read in 2*Torr, works in dyne/cm^2) c t(i) Temperature of inert in Kelvin c zc10 Initial molar fraction of condensible vapor1 (zc10+zc20=1) *------nomenclature
progname='nuetodd2o_irCH4MFC_DryCp2Up_cal07'
open(5,file='nu_CH4_MFC.dat',status='old') ! March04 Shinobu open(10,file='4pp.out',status='unknown') open(11,file='wilson.out',status='unknown') open(9,file='new4pp.out',status='unknown') open(13,file='dtemp.out',status='unknown') c open(14,file='legend3.bat',status='unknown')
205 c open(15,file='legend4.bat',status='unknown') open(7,file='upstream.out',status='unknown')
call echo
pi=3.14159d0 rg=8.3145d7 avog=6.022d23 c read two condensible species 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,*)tempcal !PP02 !RTD probe calibration added write(*,*)'tempcal = ',tempcal,' (not used)' c convert pressures to dyn/cm**2 pconv = 760.d0/1.01325d6 c read molecular weights of carrier (1), condensible (2,3) and CH4 (4) read(5,*)wmN,wm2,wm3,wm4 ! March04 Shinobu c read specific heats of gases read(5,*)cpN,cp2,cp3,cp4 ! March04 Shinobu c read latent heat, and specific heat of condensate read(5,*)dhc2,dhc3,cpc2,cpc3 c read starting value and the number of points in the output read(5,*) xstart2, ilast2 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. c istart >= 2, ifin < ilast, ni=1 (for useless roop, do k=1,ni) read(5,*)istart2, ifin2, 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 c read x values and all of the dry data p0, p(x), and the associated standard deviations. c correct the pressures i.e. baratron calibrations and pressure loss due to mesh. open(unit=4,file=dryfil,status='old') c read total number of values in dry pressure data file read(4,1)a
206 read(4,*)idend
dotncal=0.d0 p0dry=0.d0 c t0dry=273.15d0 +tempcal t0dry=0.0d0
do i=1,idend read(4,*)xd(i),po(i),deltapo(i),p(i),deltap(i),to(i),dummy2, * dummy3,dummy4,flowmain,dummy4,flowsub ! xd(i) in 0.01mm, po in 2*torr c po(i)=(po(i)*0.49967 + 2.19)-poloss !Shinobu cal04 c po(i)=po(i)/1.9888 !Shinobu cal05 c po(i)=po(i)*0.5025d0+0.457d0 !Shinobu cal07 c to(i)=to(i)*0.99147d0+0.510d0 ! RuiYang and Shinobu cal07 to(i)=to(i)*0.99375d0+0.352d0 ! Shinobu cal08 Vol.11, p.33 ccc Following equation includes the effect of poloss ccc po(i)=po(i)*0.5028d0+0.732d0 ! Shinobu cal08 Vol.11, p.54 c 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 dotncal=dotncal+(flowmain+flowsub)/idend/22.41d0 p0dry=p0dry+po(i) t0dry=t0dry+to(i) enddo close(unit=4) p0dry=p0dry/idend t0dry=t0dry/idend+273.15d0 c read the number of wet data sets and flow rate of CH4 read(5,*)ndata, dotCH4
wm0dry=(dotncal*wmN+dotCH4*wm4)/(dotncal+dotCH4) ! March04 Shinobu w40dry=dotCH4*wm4/(dotncal*wmN+dotCH4*wm4) y30dry=dotCH4/(dotncal+dotCH4) cp0dry=( dotncal*wmN*cpN+dotCH4*wm4*fcp4(t0dry,p0dry,y30dry) ) * / (dotncal*wmN+dotCH4*wm4) c write(*,*)'wm0dry,w40dry,y30dry,cp0dry=', c & wm0dry,w40dry,y30dry,cp0dry write(11,1302)dryfil
do kd = 1,ndata
read(5,*)ntype,entry1,entry2,entry3 ! March04 Shinobu
207 if(ntype.eq.0)then !pressure input (torr) pc10=entry1 pc20=entry2 tCH4=entry3 ! March04 Shinobu else if(ntype.eq.1)then !massflow and weight fraction input dotm=entry1 wfc10=entry2 wfc20=1.0d0-wfc10 c write(7,*)wfc10 tCH4=entry3 ! March04 Shinobu else write(*,*)'need to specify pressure (0)' write(*,*)'or mass flow with first weight fraction input(1)' stop end if 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. tN=0.d0 t0set=0.d0 do i=1,idenw read(4,*)xw(i),po(i),deltapo(i),p(i),deltap(i),to(i),dummy1, * dummy2,dummy3,flowmain,dummy4,flowsub !po in 2*torr c po(i)=po(i)/1.9888 !Shinobu cal05 c po(i)=po(i)*0.5025+0.457 !Shinobu cal07 ccc Following equation includes the effect of poloss ccc po(i)=po(i)*0.5028d0+0.732d0 ! Shinobu cal08 Vol.11, p.54 c t0set=t0set+to(i)/idenw c to(i)=to(i)+tempcal !ppaci02! RTD probe calibration c to(i)=to(i)*0.99147d0+0.510d0 ! RuiYang and Shinobu cal07 to(i)=to(i)*0.99375d0+0.352d0 ! Shinobu cal08 Vol.11, p.33 wet(i)=p(i)/po(i) deltawet(i)=wet(i) * *((deltap(i)/p(i))**2.0+(deltapo(i)/po(i))**2.0)**0.5 c write(*,*) xw(i), wet(i), deltawet(i) !debug tN=tN+(flowmain+flowsub)/idenw/22.41d0
208 enddo c cccccccccccccccccccccccccccccc ! Shinobu cccc read(4,*) iden_TDL do i=1,iden_TDL read(4,*) x_TDL(i),g_TDL2(i) enddo ccccccccccccccccccccccccccccccccccccccccccccc close(unit=4) c figure out the average stagnant pressure and temperature
p0=0.0 t0=0.0 do i=1,idenw p0=p0+po(i)
t0=t0+to(i) enddo p0=p0/idenw t0=t0/idenw+273.15 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'
allflux=tN+tCH4+dotm*wfc10/wm2+dotm*wfc20/wm3 wm1=(tN*wmN+tCH4*wm4)/(tN+tCH4) w40=tCH4*wm4/(tN*wmN+tCH4*wm4+dotm) y30=tCH4/allflux c !chh99 c now figure out pcondensible from calibration and average properties if(ntype.eq.1)then !now calculate pcondensible pc10=p0*dotm*wfc10/wm2/allflux ! June05 Shinobu write(*,*)'pc10= ',pc10,' torr' pc20=p0*dotm*wfc20/wm3/allflux write(*,*)'pc20= ',pc20,' torr' endif c convert pressures to dyn/cm**2 p0=p0/pconv pct0=pc10+pc20 pc10=pc10/pconv
209 pc20=pc20/pconv if((pc10+pc20).lt.1.d-18) then zc10=0.d0 else zc10=pc10/(pc10+pc20) !chh22.02.01 endif y10=pc10/p0 y20=pc20/p0 c calculate stagnation gas mass density and condensible monomer mass c density (g/cm**3) c w2,w3 are mass fraction of condensible vapor in gas wmav=(wm1*(p0-pc10-pc20)+wm2*pc10+wm3*pc20)/p0 w20=wm2*pc10/p0/wmav w30=wm3*pc20/p0/wmav wi=1.d0-w20-w30 wN0=wi-w40 c gw17-2-00 assuming vapor condenses at constant composition let's define c a fictitious mean condensible vapor molecular weight wmc if((pc10+pc20).lt.1.d-18) then wmc=0.d0 else wmc=(wm2*pc10+wm3*pc20)/(pc10+pc20) endif c also let's save the inital average molecular weight wmav0=wmav
cp0= wN0*cpN+w40*fcp4(t0,p0,y30) & +w20*fcp2(t0,p0,y10)+w30*fcp3(t0,p0,y20)
gamma=cp0dry/(cp0dry-rg*1.d-7/wm0dry) !n2 gamma gamma0=cp0/(cp0-rg*1.d-7/wmav) !initial mixture gamma rhog0=p0/rg/t0*wmav write(*,*)'wmav',' w20',' w30',' wi',' cp0',' gamma0' !chh061098 write(*,*)wmav, w20,w30,wi,cp0,gamma0 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)
210 c1 = 2.d0/(gamma-1.d0) c10 = 2.d0/(gamma0-1.d0) c2 = (gamma0+1.d0)/2.d0 c0 = (gamma0-1.d0)/gamma0 c3 = (gamma-1.d0)/gamma 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 tstt0=pstp0**c3
*********** Values at throat under Dry condition, Shinobu *************
stepm=250.0 nstepm=250 dm_1=1.d0/stepm m_1=0.d0+dm_1
t_0=t0dry p_0=p0dry
t_1=t_0 p_1=p_0 g_1=gamma cp_1=cp0dry
do i=2,nstepm
fk_1=( dgdt(t_1,p_1,0.d0,0.d0,y30dry,0.d0,0.d0,w40dry,wm0dry) & *t_1 + p_1*g_1/(g_1-1.d0)* & dgdp(t_1,p_1,0.d0,0.d0,y30dry,0.d0,0.d0,w40dry,wm0dry) )/ & ( 2.d0+(g_1-1.d0)*m_1*m_1 ) dg_1=-2.d0*fk_1*m_1*(g_1-1.d0)*2.d0*dm_1/ & (1.d0+fk_1*m_1*m_1*(g_1-1.d0)/g_1) dt_1=-t_1/(2.d0+(g_1-1.d0)*m_1*m_1)* & ( 2.d0*(g_1-1.d0)*m_1*2.d0*dm_1 + & m_1*m_1*(g_1-1.d0)/g_1*dg_1 ) dp_1=p_1/t_1*g_1/(g_1-1.d0)*dt_1 dr_1=r_1/t_1/(g_1-1.d0)*dt_1
t_2=t_0+dt_1 p_2=p_0+dp_1 r_2=r_0+dr_1 cp_2=(1.d0-w40dry)*cpN+w40dry*fcp4(t_2,p_2,y30dry) g_2=cp_2/(cp_2-rg*1.d-7/wm0dry)
t_0=t_1
211 p_0=p_1 t_1=t_2 p_1=p_2 cp_1=cp_2 g_1=g_2 m_1=m_1+dm_1 c write(*,*)'m_1,cp_1,g_1',m_1,cp_1,g_1 c write(*,*)'dg_1, dt_1, dp_1, ',dg_1,dt_1,dp_1 c write(*,*)'p_1/p0, t_1/t0',p_1/p0,t_1/t0
enddo
tstcpdry=t_1 pstcpdry=p_1
write(*,*) ' pstp0, tstt0 (for constant Cp) =', pstp0,tstt0 write(*,*) 'pstcpdry/p0dry, tstcpdry/t0dry = ', & pstcpdry/p0dry, tstcpdry/t0dry ******************************************************************** ***
pstp0dry=pstcpdry/p0dry do i=1,idend c write(*,*) i, xd(i), dry(i) !debug if((dry(i).gt. pstp0dry).and.(dry(i+1).le. pstp0dry))then c write(*,*) 'true' !debug xthroat=( pstp0dry-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 do i=1,idend x(i)=(xd(i)-xthroat)/1000.0 ! in units of cm !Shinobu for Velmex on Train B enddo c now do linear interpolation to get fixed x intervals ************************************************ ixstart=1 xstart= int(x(1)*10.d0)/10.d0 ilast=ilast2+int( (xstart2-xstart)/0.1+0.1 ) write(*,*) 'xstart= ',xstart ************************************************ c save steps in inner loop by beginning interp. where left off lasti=ixstart !chh110698
212 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 enddo
write(*,*) 'can not interpolate for point', j
5 continue enddo c we now have an array dryf(j) at fixed xs(j) intervals. now put 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) 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
213 *********** Values at throat under Wet condition, Shinobu *************
stepm=250.0 nstepm=250 dm_1=1.d0/stepm m_1=0.d0+dm_1
t_0=t0 p_0=p0 r_0=rhog0
t_1=t_0 p_1=p_0 r_1=r_0 g_1=gamma0 cp_1=cp0
do i=2,nstepm
fk_1=( dgdt(t_1,p_1,y10,y20,y30,w20,w30,w40,wmav0) & *t_1 + p_1*g_1/(g_1-1.d0)* & dgdp(t_1,p_1,y10,y20,y30,w20,w30,w40,wmav0) )/ & ( 2.d0+(g_1-1.d0)*m_1*m_1 ) dg_1=-2.d0*fk_1*m_1*(g_1-1.d0)*2.d0*dm_1/ & (1.d0+fk_1*m_1*m_1*(g_1-1.d0)/g_1) dt_1=-t_1/(2.d0+(g_1-1.d0)*m_1*m_1)* & ( 2.d0*(g_1-1.d0)*m_1*2.d0*dm_1 + & m_1*m_1*(g_1-1.d0)/g_1*dg_1 ) dp_1=p_1/t_1*g_1/(g_1-1.d0)*dt_1 dr_1=r_1/t_1/(g_1-1.d0)*dt_1
t_2=t_0+dt_1 p_2=p_0+dp_1 r_2=r_0+dr_1 cp_2= wN0*cpN+w40*fcp4(t_2,p_2,y30) & +w20*fcp2(t_2,p_2,y10)+w30*fcp3(t_2,p_2,y20) g_2=cp_2/(cp_2-rg*1.d-7/wmav)
t_0=t_1 p_0=p_1 r_0=r_1 t_1=t_2 p_1=p_2 r_1=r_2 cp_1=cp_2 g_1=g_2 m_1=m_1+dm_1 c write(*,*)'m_1,cp_1,g_1',m_1,cp_1,g_1 c write(*,*)'dg_1, dt_1, dp_1, ',dg_1,dt_1,dp_1
214 c write(*,*)'p_1/p0, t_1/t0',p_1/p0,t_1/t0
enddo
gammam=g_1 tstarcp=t_1 pstarcp=p_1 rstarcp=r_1 ustarcp=dsqrt(gammam*rg*tstarcp/wmav)
write(*,*)'p*/p0, t*/t0, r*/r0 (constant Cp) ', & pstp0w,1.d0/c2,c2**erho write(*,*)'p_1/p0, t_1/t0, r_1/r0 ',p_1/p0,t_1/t0,r_1/rhog0 write(*,*) 'gamma0, gammam',gamma0,gammam ******************************************************************** ***
do i=1,idenw c write(*,*) i, xw(i), wet(i) !debug if((wet(i).gt.pstarcp/p0).and.(wet(i+1).le.pstarcp/p0))then c write(*,*) 'true' !debug xthroat=xw(i)+(pstarcp/p0-wet(i))/ & (wet(i+1)-wet(i))*(xw(i+1)-xw(i)) go to 5002 endif enddo 5002 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. !chh110698 ixstart=0 !chh110698 do i=1,idenw x(i)=(xw(i)-xthroat)/1000.0 ! in units of cm if(x(i).le.xstart)ixstart=i !chh110698 enddo c write(*,*) 'ixstart= ',ixstart !chh110698 write(*,*) 'throat shifted' !debug c now do linear interpolation to get fixed x intervals lasti=ixstart !chh110698 do j=1,ilast c xs values have already been assigned in dry data analysis c write(*,*) xs(j) !debug do i=lasti,idenw !chh110698
215 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 !chh110698 goto 6 endif enddo
write(*,*) 'can not interpolate for point', j 6 continue enddo c cccccccccccccccccccccccccccccccccc ! Shinobu lasti=1 do j=1,ilast c write(*,*) xs(j) !debug do i=lasti,iden_TDL if((x_TDL(i).le.xs(j)).and.(x_TDL(i+1).gt.xs(j))) then g_TDL3(j)=g_TDL2(i)+(xs(j)-x_TDL(i))* & (g_TDL2(i+1)-g_TDL2(i))/(x_TDL(i+1)-x_TDL(i)) lasti=i !chh110698 goto 62 endif enddo
write(*,*) 'can not interpolate for point', j 62 continue enddo ccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccc c we now have an array wetf(j) at fixed xs(j) intervals. now put 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
216 call smooth(md,nd,i,k0,sval,wetf) swet(i)=sval enddo write(*,*) 'finished interpolating points' 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'
********************************************** ifin=ifin2+int( (xstart2-xstart)/0.1+0.1 ) istart0=1 **********************************************
npts=ifin-istart0+1 nnpts=(npts-1)*nint+1 jinit=nnpts+2*nint+istart0-1 do i=ifin+1,istart0,-1 delx=xs(i)-xs(i-1) delprd = sdry(i)-sdry(i-1) c delprwi = sweti(i)-sweti(i-1) delprw = swet(i)-swet(i-1) delg=g_TDL3(i) - g_TDL3(i-1) ! Shinobu 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 if(dabs(xs(j)).LT.1.d-4) ithroat=j pp0d(j) = sdry(i)-delprd*fint c pp0i(j) = sweti(i)-delprwi*fint pp0(j) = swet(i)-delprw *fint g_TDL(j)=g_TDL3(i)-delg*fint ! Shinobu jp=jp+1 enddo enddo ifin1=istart0+nnpts-1
***************************************************************** istart= istart0+ & int( (xstart2-xstart)/0.1+istart2-istart0+0.1 )*nint write(*,*)'ithroat =',ithroat *****************************************************************
*** Pressure and temperature upstream of the integration region ****
217 t_0=tstarcp p_0=pstarcp
cp_0= wN0*cpN+w40*fcp4(t_0,p_0,y30) & +w20*fcp2(t_0,p_0,y10)+w30*fcp3(t_0,p_0,y20) g_0=cp_0/(cp_0-rg*1.d-7/wmav)
dp=( pp0(ithroat+1)-pp0(ithroat-1) )*p0 dt=t_0/p_0*(g_0-1.d0)/g_0*dp
t_is_up(ithroat)=tstarcp t_is_up(ithroat+1)=tstarcp+dt/2.d0 t_is_up(ithroat-1)=tstarcp-dt/2.d0
t_1=t_is_up(ithroat+1) p_1=pp0(ithroat+1)*p0
do i=ithroat+2,istart cp_1= wN0*cpN+w40*fcp4(t_1,p_1,y30) & +w20*fcp2(t_1,p_1,y10)+w30*fcp3(t_1,p_1,y20) g_1=cp_1/(cp_1-rg*1.d-7/wmav)
dp=( pp0(i)-pp0(i-2) )*p0 dt=t_1/p_1*(g_1-1.d0)/g_1*dp
t_is_up(i)=t_is_up(i-2)+dt
t_1=t_is_up(i) p_1=pp0(i)*p0 enddo
t_1=t_is_up(ithroat-1) p_1=pp0(ithroat-1)*p0
do i=ithroat-2,istart0,-1 cp_1= wN0*cpN+w40*fcp4(t_1,p_1,y30) & +w20*fcp2(t_1,p_1,y10)+w30*fcp3(t_1,p_1,y20) g_1=cp_1/(cp_1-rg*1.d-7/wmav)
dp=( pp0(i)-pp0(i+2) )*p0 dt=t_1/p_1*(g_1-1.d0)/g_1*dp
t_is_up(i)=t_is_up(i+2)+dt
t_1=t_is_up(i) p_1=pp0(i)*p0 enddo
write(7,700) 700 format(' x(cm) pp0 p(Torr) Tisw(K)')
218 do i=istart0,istart write(7,710) xs(i),pp0(i),pp0(i)*p0*pconv,t_is_up(i) 710 format(f7.2,f8.4,2f10.2) enddo close(unit=7) ******************************************************************** ***
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
***********Values at the start point of integration for Dry, Shinobu **********
stepp=100.0 nstepp=100 dp_1=(pp0d(istart-1)*p0dry-pstcpdry)/stepp t_0=tstcpdry p_0=pstcpdry cp_0=(1.d0-w40dry)*cpN+w40dry*fcp4(t_0,p_0,y30dry) g_0=cp_0/(cp_0-rg*1.d-7/wm0dry) m_0=1.d0 fk_0=( dgdt(t_0,p_0,0.d0,0.d0,y30dry,0.d0,0.d0,w40dry,wm0dry) & *t_0 + p_0*g_0/(g_0-1.d0)* & dgdp(t_0,p_0,0.d0,0.d0,y30dry,0.d0,0.d0,w40dry,wm0dry) )/ & ( 2.d0+(g_0-1.d0)*m_0*m_0 ) a_0=1.d0
dt_0=t_0/p_0*(g_0-1.d0)/g_0*dp_1 dm_0= -(2.d0+(g_0-1.d0)*m_0**2) & *( g_0+fk_0*(m_0**2)*(g_0-1.d0) ) & /(2.d0*(g_0**2)*m_0)/p_0*dp_1 da_0= -a_0*(m_0**2-1.d0)/(g_0*m_0**2)/p_0*dp_1
t_1=t_0+dt_0 m_1=m_0+dm_0 p_1=p_0+dp_1 a_1=a_0+da_0 cp_1=(1.d0-w40dry)*cpN+w40dry*fcp4(t_1,p_1,y30dry) g_1=cp_1/(cp_1-rg*1.d-7/wm0dry) fk_1=( dgdt(t_1,p_1,0.d0,0.d0,y30dry,0.d0,0.d0,w40dry,wm0dry) & *t_1 + p_1*g_1/(g_1-1.d0)* & dgdp(t_1,p_1,0.d0,0.d0,y30dry,0.d0,0.d0,w40dry,wm0dry) )/
219 & ( 2.d0+(g_1-1.d0)*m_1*m_1 )
do i=2, nstepp dt_1=t_1/p_1*(g_1-1.d0)/g_1*2.d0*dp_1 dm_1= -(2.d0+(g_1-1.d0)*m_1**2) & *( g_1+fk_1*(m_1**2)*(g_1-1.d0) ) & /(2.d0*(g_1**2)*m_1)/p_1*2.d0*dp_1 da_1= -a_1*(m_1**2-1.d0)/(g_1*m_1**2)/p_1*2.d0*dp_1
t_2=t_0+dt_1 m_2=m_0+dm_1 a_2=a_0+da_1 p_2=p_0+2.d0*dp_1
cp_2=(1.d0-w40dry)*cpN+w40dry*fcp4(t_2,p_2,y30dry) g_2=cp_2/(cp_2-rg*1.d-7/wm0dry) fk_2=( dgdt(t_2,p_2,0.d0,0.d0,y30dry,0.d0,0.d0,w40dry,wm0dry) & *t_2 + p_2*g_2/(g_2-1.d0)* & dgdp(t_2,p_2,0.d0,0.d0,y30dry,0.d0,0.d0,w40dry,wm0dry) )/ & ( 2.d0+(g_2-1.d0)*m_2*m_2 )
t_0=t_1 m_0=m_1 a_0=a_1 p_0=p_1 t_1=t_2 m_1=m_2 a_1=a_2 g_1=g_2 fk_1=fk_2 p_1=p_2 c write(*,*)'dt_1,dm_1,da_1 =',dt_1,dm_1,da_1
enddo
tisd(istart-1)=t_1 aratio(istart-1)=a_1 ar_TDL(istart-1)=a_1 !harshad mdry(istart-1)=m_1
dp_2=dp_1+(pp0d(istart)-pp0d(istart-1))*p0dry dt_1=t_1/p_1*(g_1-1.d0)/g_1*dp_2 dm_1= -(2.d0+(g_1-1.d0)*m_1**2) & *( g_1+fk_1*(m_1**2)*(g_1-1.d0) ) & /(2.d0*(g_1**2)*m_1)/p_1*dp_2 da_1= -a_1*(m_1**2-1.d0)/(g_1*m_1**2)/p_1*dp_2
t_2=t_0+dt_1
220 m_2=m_0+dm_1 a_2=a_0+da_1 p_2=p_0+dp_2
tisd(istart)=t_2 mdry(istart)=m_2 aratio(istart)=a_2 ar_TDL(istart)=a_2 !harshad ******************************************************************** ************* 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. c msqw = c10*((1.d0/pp0(istart-1))**c0-1.d0) c tt0(istart-1)=1.d0/(1.d0+msqw/c10) c rr0(istart-1)=(1.d0+msqw/c10)**erho c msqw = c10*((1.d0/pp0( istart))**c0-1.d0) c tt0(istart)=1.d0/(1.d0+msqw/c10) c rr0(istart)=(1.d0+msqw/c10)**erho
***********Values at the start point of integration for Wet, Shinobu **********
stepp=100.0 nstepp=100 dp_1=(pp0(istart-1)*p0-pstarcp)/stepp
t_0=tstarcp p_0=pstarcp r_0=rstarcp g_0=gammam
dt_0=t_0/p_0*(g_0-1.d0)/g_0*dp_1 dr_0=r_0/p_0/g_0*dp_1 t_1=t_0+dt_0 r_1=r_0+dr_0 p_1=p_0+dp_1 cp_1= wN0*cpN+w40*fcp4(t_1,p_1,y30) & +w20*fcp2(t_1,p_1,y10)+w30*fcp3(t_1,p_1,y20) g_1=cp_1/(cp_1-rg*1.d-7/wmav)
do i=2,nstepp
dt_1=t_1/p_1*(g_1-1.d0)/g_1*2.d0*dp_1 dr_1=r_1/p_1/g_1*2.d0*dp_1 t_2=t_0+dt_1
221 r_2=r_0+dr_1 p_2=p_0+2.d0*dp_1 cp_2= wN0*cpN+w40*fcp4(t_2,p_2,y30) & +w20*fcp2(t_2,p_2,y10)+w30*fcp3(t_2,p_2,y20) g_2=cp_2/(cp_2-rg*1.d-7/wmav)
t_0=t_1 r_0=r_1 p_0=p_1 t_1=t_2 r_1=r_2 cp_1=cp_2 g_1=g_2 p_1=p_2
enddo
tt0_is(istart-1)=t_1/t0 rr0_is(istart-1)=r_1/rhog0 pp0_is(istart-1)=p_1/p0
msqw = c10*((1.d0/pp0(istart-1))**c0-1.d0) write(*,*)'pp0(istart-1), tt0, rr0', & pp0(istart-1),1.d0/(1.d0+msqw/c10),(1.d0+msqw/c10)**erho write(*,*)'p_1/p0,t_1/t0,r_1/rhog0', & pp0_is(istart-1),tt0_is(istart-1),rr0_is(istart-1)
dp_2=dp_1+(pp0(istart)-pp0(istart-1))*p0 dt_1=t_1/p_1*(g_1-1.d0)/g_1*dp_2 dr_1=r_1/p_1/g_1*dp_2 t_2=t_0+dt_1 r_2=r_0+dr_1
tt0_is(istart)=t_2/t0 rr0_is(istart)=r_2/rhog0 pp0_is(istart)=(p_0+dp_2)/p0
msqw = c10*((1.d0/pp0( istart))**c0-1.d0) write(*,*)'pp0(istart), tt0, rr0 ', & pp0(istart),1.d0/(1.d0+msqw/c10),(1.d0+msqw/c10)**erho write(*,*)'p_2/p0,t_2/t0,r_2/rhog0', & pp0_is(istart),tt0_is(istart),rr0_is(istart)
tt0(istart-1)=tt0_is(istart-1) tt0_TDL(istart-1)=tt0_is(istart-1) !harshad rr0(istart-1)=rr0_is(istart-1) rr0_TDL(istart-1)=rr0_is(istart-1) !harshad tt0(istart)=tt0_is(istart) tt0_TDL(istart)=tt0_is(istart) !harshad rr0(istart)=rr0_is(istart)
222 rr0_TDL(istart)=rr0_is(istart) !harshad ******************************************************************** ***
g(istart)=0.d0 g(istart-1)=0.d0 fc(istart)=0.0d0 !fraction condensed fc_TDL(istart)=0.0d0 !harshad write(10,1024) progname 1024 format('Program: ',a60)
write(10,1011)p0*pconv,devp0,t0-273.15, t0set, rhog0*1.0d3 !4pp plots write(10,1010) dotm,specie(1),wfc10 !4pp plots 1010 format('Weight flux of condensable =',f6.2, & ' g/min , Fraction of ',a, '=',f7.3) !4pp plots 1011 format('p0= ',f6.2,'+/-',f4.2,' Torr T0=',f6.2, & ' C (set T0=',f6.2,') rho0=', e11.4,' kg/m3') !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) allflux, (dotncal+dotCH4) !4pp plots 1013 format( 'Total mole flux =', & f6.3,' mol/min (',f6.3,' in Dry )' ) !4pp plots write(10,1019) tCH4, dotCH4 !4pp plots 1019 format( ' CH4 mole flux =', & f6.3,' mol/min (',f6.3,' in Dry )' ) !4pp plots write(10,1014)wetfil,dryfil !4pp plots 1014 format('@subtitle \"',a30,'with dry trace ',a30,'\"') !4pp plots write(10,1015) !4pp plots 1015 format(' x(cm) u(m/s) T(K) p/p0 Tis p/p0_is', &' MoleFract. g g/g_inf A/A* r/r0', &' Tisd p/p0_isd') !4pp plots c write(12,1016)kd,pc10*pconv,specie(1),pc20*pconv, c +specie(2) !4pp plots c 1016 format('legend string ',i2,' \"',2(f7.4,'torr ',a),'\"') !4pp plots c write(14,1017)kd, wetfil !4pp plots c 1017 format('legend string ',i2,' \"',a13,'\"') !4pp plots c write(15,1018)kd-1,p0*pconv,devp0,t0-273.15 !4pp plots c 1018 format('legend string ',i2,' \"',f6.2,'+/-',f4.2,'torr ', c & f6.2,'celsius"') !4pp plots
223 write(9,1037) !4pp plots in new output file 1037 format(' x(cm) u_PTM T_PTM g_PTM T_is Tptm-Tis P/po_is', &' P/po g/g_inf T_tdl A/A*_PTM A/A*_TDL gginf_TDL', &' r/r0_TDL r/r0 g_TDL u_TDL')
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
*********** Integration of the isentropic curve for Dry, Shinobu **************
dp_dry=( pp0d(i+1)-pp0d(i-1) )*p0dry/2.d0
p_dry=pp0d(i)*p0dry cp_dry=(1.d0-w40dry)*cpN + w40dry*fcp4(tisd(i),p_dry,y30dry) g_dry=cp_dry/(cp_dry-rg*1.d-7/wm0dry) fk_dry=( dgdt(tisd(i),p_dry,0.d0,0.d0,y30dry, & 0.d0,0.d0,w40dry,wm0dry) & *tisd(i) + p_dry*g_dry/(g_dry-1.d0)* & dgdp(tisd(i),p_dry,0.d0,0.d0,y30dry, & 0.d0,0.d0,w40dry,wm0dry) )/ & ( 2.d0+(g_dry-1.d0)*mdry(i)**2 )
dt_dry=tisd(i)/p_dry*(g_dry-1.d0)/g_dry*dp_dry dm_dry= -( 2.d0+(g_dry-1.d0)*mdry(i)**2 ) & *( g_dry+fk_dry*(mdry(i)**2)*(g_dry-1.d0) ) & /(2.d0*(g_dry**2)*mdry(i))/p_dry*dp_dry da_dry= -aratio(i)*(mdry(i)**2-1.d0)/(g_dry*mdry(i)**2)/ & p_dry*dp_dry
tisd(i+1)=tisd(i-1)+2.d0*dt_dry mdry(i+1)=mdry(i-1)+2.d0*dm_dry aratio(i+1)=aratio(i-1)+2.d0*da_dry
dar=dlog(aratio(i+1)/aratio(i-1))/2.d0 c write(*,*) 'Integration of the isentropic curve for Dry OK' ******************************************************************** ************* ccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccc c Arearatio for constant Cp of CH4
224 c c msq = c1*((1.d0/pp0d(i))**c3-1.d0) c aratio(i)= dsqrt(((c1/eam2*(1.d0+msq/c1))**eam2)/msq) c msq = c1*((1.d0/pp0d(i+1))**c3-1.d0) c aratio(i+1) = dsqrt(((c1/eam2*(1.d0+msq/c1))**eam2)/msq) c dar=dlog(aratio(i+1)/aratio(i-1))/2.d0 c c cccccccccccccccccccccccccccccccccccccccccccccccccccccccccccc
*********** Integration of the isentropic curve for Wet, Shinobu **************
dp=(pp0(i+1)-pp0(i-1))/2.d0
t_is(i)=tt0_is(i)*t0
u_is=ustarcp*rstarcp/rhog0/rr0_is(i)/aratio(i) mssq_is=(u_is/ustarcp)**2 cp_is= wN0*cpN+w40*fcp4(t_is(i),pp0_is(i)*p0,y30) & +w20*fcp2(t_is(i),pp0_is(i)*p0,y10) & +w30*fcp3(t_is(i),pp0_is(i)*p0,y20) cpr_is=cp_is/cp0 hpara_is=1.d0-(gamma0-1.d0)/gamma0/cpr_is
tempA=(1.d0-t_is(i)/tstarcp/gammam/mssq_is)/rr0_is(i) tempG=t0/tstarcp/gammam/mssq_is tempJ=rr0_is(i)*tt0_is(i)/ & (hpara_is-t_is(i)/tstarcp/gammam/mssq_is)
dtt0_is=(tt0_is(i)-tempA*tempJ)*dar dpp0_is=-tempJ*dar drr0_is=-(rr0_is(i)+tempG*tempJ)*dar
tt0_is(i+1)=tt0_is(i-1)+2.d0*dtt0_is pp0_is(i+1)=pp0_is(i-1)+2.d0*dpp0_is rr0_is(i+1)=rr0_is(i-1)+2.d0*drr0_is c write(*,*) 'Integration of the isentropic curve for Wet OK' *************** Smoothing, Shinobu ************************************ t_is_s(i)=t0*( tt0_is(i-1)+2.d0*tt0_is(i)+tt0_is(i+1) )/4.d0 pp0_is_s(i)=( pp0_is(i-1)+2.d0*pp0_is(i)+pp0_is(i+1) )/4.d0 c t_is_s(i)=t0*tt0_is(i) c pp0_is_s(i)=pp0_is(i) ******************************************************************** *** c write(*,*)'mssq,mssq_is',mssq,mssq_is c write(*,*)'cp,cp_is',cp,cp_is c write(*,*)'hpara,hpara_is',hpara,hpara_is
225 c write(*,*)'gamma0,gammam',gamma0,gammam c write(*,*)'tempA, tempJ,tempG',tempA,tempJ,tempG c write(*,*) rr0_is(i) c write(*,*)'dar',dar c write(*,*)'dtt0,dpp0',dtt0_is,dpp0_is c write(*,*)'mdry(i), Mach ,fk_dry=', c & mdry(i), dsqrt(c1*((1.d0/pp0d(i))**c3- 1.d0)),fk_dry
******************************************************************** *** t(i)=tt0(i)*t0 t_TDL(i)=tt0_TDL(i)*t0 !harshad u(i)=ustarcp*rstarcp/rhog0/rr0(i)/aratio(i) u_TDL(i)=ustarcp*rstarcp/rhog0/rr0_TDL(i)/ar_TDL(i) ! harshad mssq=(u(i)/ustarcp)**2 mssq_TDL=(u_TDL(i)/ustarcp)**2 c fcon=dotm * (wfc10/wm2 + (1-wfc10)/wm3) c if((pc10+pc20).lt.1.d-18) then c y1=0.d0 c y2=0.d0 c else c y1=(1-fc(i))*fcon/(allflux-fc(i)*fcon)*y10/(y10+y20) c y2=(1-fc(i))*fcon/(allflux-fc(i)*fcon)*y20/(y10+y20) c endif c y3=y30*allflux/(allflux-fc(i)*fcon) c write(*,*)'(y1+y2+y3+tN/allflux*y3/y30), (wN0+w20+w30+w40)', c & y1+y2+y3+tN/allflux*y3/y30, wN0+w20+w30+w40 c harshad-y values required only in case of clustering y1=0 y1_TDL=0 y2=0 y2_TDL=0 y3=0 y3_TDL=0 c gw2-17-00 update specific heat if((pc10+pc20).lt.1.d-18) then cpv=0.d0 cpc=0.d0 else cpv=( w20*fcp2(t(i),pp0(i)*p0,y1) + & w30*fcp3(t(i),pp0(i)*p0,y2) )/(w20+w30) cpc=( w20*fcpl2(t(i))+w30*fcpl3(t(i)) )/(w20+w30) endif cp= wN0*cpN+w40*fcp4(t(i),pp0(i)*p0,y3) & +(w20+w30-g(i))*cpv+g(i)*cpc
226 cpr=cp/cp0 c c gw2-17-00 update specific heat for TDL if((pc10+pc20).lt.1.d-18) then cpv_TDL=0.d0 cpc_TDL=0.d0 else cpv_TDL=( w20*fcp2(t_TDL(i),pp0(i)*p0,y1_TDL) + & w30*fcp3(t_TDL(i),pp0(i)*p0,y2_TDL) )/(w20+w30) cpc_TDL=( w20*fcpl2(t_TDL(i))+w30*fcpl3(t_TDL(i)) )/(w20+w30) endif cp_TDL= wN0*cpN+w40*fcp4(t_TDL(i),pp0(i)*p0,y3_TDL) & +(w20+w30-g_TDL(i))*cpv_TDL+g_TDL(i)*cpc_TDL cpr_TDL=cp_TDL/cp0 c gw2-17-00 update "mu/(1-g)" = wmu, and related factors if((pc10+pc20).lt.1.d-18) then wmu=wm1 wg(i)=0.d0 else wmu=wm1*wmc/(wi*wmc+(w20+w30-g(i))*wm1) wg(i)=wmu/(wmc) endif wmuu0=wmu/wmav0 c c gw2-17-00 update "mu/(1-g)" = wmu_TDL, and related factors if((pc10+pc20).lt.1.d-18) then wmu_TDL=wm1 wg_TDL=0.d0 else wmu_TDL=wm1*wmc/(wi*wmc+(w20+w30-g_TDL(i))*wm1) wg_TDL=wmu_TDL/(wmc) endif wmuu0_TDL=wmu_TDL/wmav0
hpara=wmuu0-(gamma0-1.d0)/gamma0/cpr dr=dp/tstarcp*t0/gammam/mssq-rr0(i)*dar dgp=(hpara-t(i)/tstarcp/mssq/gammam)/rr0(i)*dp+tt0(i)*dar if((pc10+pc20).lt.1.d-18) then dg=0.d0 else dg=dgp*cp*t0/(fdhc(wfc10,t(i))-cp*t(i)*wg(i)) ! Shinobu endif c gw2-17-00 update dtt0 dtt0=(wmuu0-t(i)/tstarcp/gammam/mssq)/rr0(i)*dp+ & tt0(i)*(dar+wg(i)*dg)
227 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
if((w20+w30).gt.0.0)then fc(i+1)=g(i+1)/(w20+w30) ! March04 Shinobu else fc(i+1)=0.0 end if c c write(*,*) 'Integration of the Wet trace OK' cc ***********************correction for the code when g is input******** ap_TDL=wmuu0_TDL*gamma0-(gamma0-1.d0)/cpr_TDL ! Shinobu h_TDL=ap_TDL/gamma0 ! Shinobu cc ccccccccccccccccccc ! Shinobu cccccccccccccccccccccccc
dg_TDL=(g_TDL(i+1)-g_TDL(i-1))/2.0d0 tempA=(wmuu0_TDL-t_TDL(i)/tstarcp/gammam/mssq_TDL)/rr0_TDL(i) tempB=tt0_TDL(i) tempC=tt0_TDL(i)*wg_TDL tempF=fdhc(wfc10,t_TDL(i))/cp_TDL/t0-tt0_TDL(i)*wg_TDL tempD=(h_TDL - t_TDL(i)/tstarcp/gammam/mssq_TDL)/rr0_TDL(i)/tempF tempE=tt0_TDL(i)/tempF dlar_TDL=dg_TDL/tempE-tempD/tempE*dp dtt0_TDL=tempA*dp+tempB*dlar_TDL+tempC*dg_TDL dr_TDL=dp/tstarcp*t0/gammam/mssq_TDL-rr0_TDL(i)*dlar_TDL
ar_TDL(i+1)=ar_TDL(i-1)*dexp(2.0d0*dlar_TDL) rr0_TDL(i+1)=rr0_TDL(i-1)+2.0d0*dr_TDL tt0_TDL(i+1)=tt0_TDL(i-1)+2.0d0*dtt0_TDL wm_TDL=wmav0*dotncal/ & (dotncal-(dotncal-tN-tCH4)*fc_TDL(i)) ! Molecular weight for c c sound velocity ga_TDL=cp_TDL/(cp_TDL-8.3145/wm_TDL) ! Specific heat ratio for c sound velocity a_TDL=(ga_TDL*8.3145*t_TDL(i)/wm_TDL*1000.0)**0.5
if((w20+w30).gt.0.0)then fc_TDL(i+1)=g_TDL(i+1)/(w20+w30) ! Shinobu else fc_TDL(i+1)=0.0 end if write(9,1105)xs(i),u(i)/100,t(i),g(i),t_is_s(i),t(i)- t_is_s(i), & pp0_is_s(i),pp0(i),fc(i), & t_TDL(i),aratio(i),ar_TDL(i),fc_TDL(i),rr0_TDL(i), & rr0(i),g_TDL(i),u_TDL(i)/100
228 1105 format(3f8.2,f8.4,2f8.2,3f8.4,f8.2, & 5f8.4,f8.4,f8.2) cc ***********************correction for the code when g is input over******** write(10,1020)xs(i),u(i)/100,t(i),pp0(i),t_is_s(i),pp0_is_s(i), ! June05 Shinobu * (1-fc(i))*fcon/(allflux-fc(i)*fcon),g(i),fc(i),aratio(i), * rr0(i),tisd(i),pp0d(i)
1020 format(f8.3,f10.2,f8.2,f8.4,f8.2,f8.4,2e13.4,f8.4,f8.4,f8.4, & f8.2,f8.4) !4pp plots 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)-t_is_s(i) and t(i)-tisd
do i = istart,ifin1-1 dtemp(i,kd) = t(i)-t_is_s(i) dt1 = t(i) - t_is_s(i) dt2 = t(i+1) -t_is_s(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 = pp0_is_s(i)+(0.5-dt1)/(dt2-dt1)*(pp0_is_s(i+1)- pp0(i)) ton = t(i)+(0.5-dt1)/(dt2-dt1)*(t(i+1)-t(i)) tiswon = t_is_s(i)+(0.5-dt1)/(dt2-dt1)*(t_is_s(i+1)- t_is_s(i)) else endif enddo write(*,*)'using the t-t_is_s = 0.5 k' write(*,1300)xon,pp0on*pct0,ton, & pp0ion*pct0,tiswon 1300 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,
& pp0on*pc10*pconv,pp0on*pc20*pconv,wetfil,xon 1301 format(f8.2,f8.2,f8.4,f8.2,f8.4,f8.4,f8.4,2x,a13,f7.1) 1302 format('@\"t0 p0 pct ton pon p1on p2on', & 6x,a13,'\"') xon=0.0 pp0on=0.0 ton=0.0 do i = istart,ifin1-1
229 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)) pp0don = pp0d(i)+(0.5-dt1)/(dt2-dt1)*(pp0d(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)) else endif enddo write(*,*)'using the criterion t-tisd = 0.5 k' write(*,1300)xon,pp0on*(pc10+pc20)*pconv,ton, & pp0don*(pc10+pc20)*pconv,tisdon
write(*,*) 'finished integration' enddo enddo c now write out the dtemp files to dtemp.out
do i = istart,ifin1 write(13,1313)xs(i),(dtemp(i,j),j=1,ndata) enddo 1313 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 c regardless of the spacing c c m - order of the highest polynomial used in smoothing c n - number of y points in interval over which smoothing is performed c k - point whose smoothed value is desired c k0 - first point in set of n c sval - smoothed value returned to calling program c 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.
230 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(9,2)a goto 15 99 continue rewind 5 return 1 format(a100) 2 format(1x,a100) 3 format(1h1,20x,'input file',//) end *------chh22.02.01---* * c real function fdhc(dhc) c fdhc = dhc c return * real*8 function fdhc(wfc10,tk) double precision zc10,wfc10,tk,rg
231 *------general nomenclature-* c rg universal gas constant in units of c tk temperature of vapor condensing in kelvin c zc10 molar fraction of condensible 1 in vapor (zc10+zc20=1.0) *------condensible nomenclature-* c a2h2o - a4h2o h2o vapor pressure constants, wagner correlation c a1d2o - a6d2o d2o vapor pressure constants, c mwd2o d2o molecular weight c z d2o intermediate variable *------* double precision a1d2o,a2d2o,a3d2o,a4d2o,a5d2o,a6d2o,z,mwd2o !d2o pve constants double precision bbu,cbu,mwetod,dhcetod
rg=8.3145d0
*------* *-d2o clausius-clapeyron relation applied to equilibrium vapor pressure *-d2o valid for temperature range of 275-823K *-d2o hill, mcmillan, and lee, j. phys chem ref data, vol 11, no.1, p1-14 (1982) a1d2o= -7.81583d0 a2d2o= 17.6012d0 a3d2o=-18.1747d0 a4d2o= -3.92488d0 a5d2o= 4.19174d0 a6d2o=643.89d0 mwd2o=20.03d0
z=1-tk/a6d2o d2oa=a1d2o*z+a2d2o*z**1.9+a3d2o*z**2+a4d2o*z**5.5+a5d2o*z**10. d2ob=a1d2o+1.9d0*a2d2o*z**0.9+2.d0*a3d2o*z+5.5d0*a4d2o*z**4.5 &+10.d0*a5d2o*z**9. dhcd2o=-rg*(a6d2o*d2oa+tk*d2ob)/mwd2o
*------* *-BuOH clausius-clapeyron relation applied to equilibrium vapor pressure *-BuOH valid for temperature range of 243.2-303.2 K * And T. Schmeling and R. Strey, Ber. Bunsenges. Phys. Chem., vol 87, p871-874 (1983)
232 c bbu= 9412.61d0 c cbu= 10.54d0 c mwbuOH=74.12d0 c dhcbuOH=rg*(bbu-cbu*tk)/mwbuOH c Use corrected equation 20 of Ruzicka and Majer J physical chem ref data 23, 1994 p 1-39 c Note the T is missing from the a1 term!Original units are J/mol a0 = 2.94690d0 a1 = -2.051933d-3 a2 = 1.903683d-6 Tb = 423.932d0 mwetod=128.260d0
Term1 = rg*exp(a0 + a1*tk + a2*tk*tk)
Term2 = Tb + tk*(tk - Tb)*(a1+2.0d0*a2*tk) ccc dhcetod= Hvap von Nonane!!!
dhcetod = (Term1*Term2)/mwetod
*------* fdhc = (wfc10*dhcetod)+(1.d0-wfc10)*dhcd2o c write(28,*)'dhc debug: fdhc= ',fdhc,' wfc10= ',wfc10,' tk= ',tk,' K' !debug dhc return
end
*------*
***************** Functions for Cp **********************************
real*8 function fcp2(tk,p,y1) double precision tk,p,y1 double precision mw,a0,a1,a2 c** Cp of BuOH ****** p: Total static pressure ****** y1: Mole fraction of condensable 1 in vapor phase c mw=74.12 c a0=30.941d0 c a1=0.10037d0 c a2=7.322d-5 c fcp2=(a0+a1*tk+a2*tk*tk)/mw (data for EtOD)
233 c fcp2=1.473 used a const cp for BuOh?? c [J/g*K]
*----Cp Nonane--Ruzicka & Majer J.Phys.Chem.Ref.Data.Vol.23.No.1,1994 ------***NEU machen * taken from Bures et al, Chem Eng Sci, 36, 529-537 (1981) c------Einheit J/molK------
A = 148.15036d0 B1 = 288.24904d0 B2 = 178.57491d0 C1 = 1380.8003d0 C2 = 3051.1566d0
Term1 = (C1/tk)*(C1/tk)*dexp(-C1/tk)/(1.0d0 - dexp(- C1/tk))**2
Term2 = (C2/tk)*(C2/tk)*dexp(-C2/tk)/(1.0d0 - dexp(- C2/tk))**2
cpnonane = A + B1*Term1 + B2*Term2
mw=128.260d0
fcp2 = cpnonane/mw c unit for the code should be J/g*K
return end
real*8 function fcp3(tk,p,y2) double precision tk,p,y2 double precision rg,mw,a0,a1,a2,a3 c** Cp of D2O: Fitted to the data in JCP 22, 2051 (1954) at T= 160- 340 K rg=8.3145 mw=20.03 a0=4.1712 a1=-2.2388d-3 a2= 8.6096d-6 a3=-5.6304d-9 fcp3=rg*(a0+a1*tk+a2*tk**2+a3*tk**3)/mw c fcp3=1.710d0 return end
real*8 function fcp4(tk,p,y3)
234 double precision tk,p,y3 double precision rg,mw,a0,a1,a2,a3 c** Cp of CH4: Fitted to the data in J. Chem. Eng. Data 8, 547 (1963) at T= 160-340 K rg=8.3145 mw=16.04 a0=4.337 a1=-3.7677d-3 a2= 9.1107d-6 a3= 1.0178d-9 fcp4=rg*(a0+a1*tk+a2*tk**2+a3*tk**3)/mw c fcp4=2.226d0 return end
************** Temperature derivative of Cp ****************
real*8 function fdcp2dt(tk,p,y1) double precision tk,p,y1,mw,a1,a2,c1,c2 c mw=74.12 c a1=0.10037d0 c a2=7.322d-5 c fdcp2dt=(a1+2.d0*a2*tk)/mw c----Cp Nonane--Ruzicka & Majer J.Phys.Chem.Ref.Data.Vol.23.No.1,1994 ------***NEU machen * taken from Bures et al, Chem Eng Sci, 36, 529-537 (1981) c------Einheit J/molK------
A = 148.15036d0 B1 = 288.24904d0 B2 = 178.57491d0 C1 = 1380.8003d0 C2 = 3051.1566d0 mwnonane=128.258d0 ccc Näherung für cp/dt
tk1 = tk+ 0.01d0 tk2 = tk - 0.01d0
Term1 = (C1/tk1)*(C1/tk1)*dexp(-C1/tk1)/ & (1.0d0 - dexp(-c1/tk1))**2 Term2 = (C2/tk1)*(C2/tk1)*dexp(-C2/tk1)/ & (1.0d0 - dexp(-c2/tk1))**2
cpnonane1 = (A + B1*Term1 + B2*Term2)/mwnonane
Term1 = (C1/tk2)*(C1/tk2)*dexp(-C1/tk2)/ & (1.0d0 - dexp(-c1/tk2))**2 Term2 = (C2/tk2)*(C2/tk2)*dexp(-C2/tk2)/ & (1.0d0 - dexp(-c2/tk2))**2
235 cpnonane2 = (A + B1*Term1 + B2*Term2)/mwnonane
fdcp2dt = (cpnonane1-cpnonane2)/(0.02d0)
return end
real*8 function fdcp3dt(tk,p,y2) double precision tk,p,y2 double precision rg,mw,a1,a2,a3 c** Cp of D2O: Fitted to the data in JCP 22, 2051 (1954) at T= 160- 340 K rg=8.3145 mw=20.03 a1=-2.2388d-3 a2= 8.6096d-6 a3=-5.6304d-9 fdcp3dt=rg*(a1+2.d0*a2*tk+3.d0*a3*tk**2)/mw c fdcp3dt=0.0 return end
real*8 function fdcp4dt(tk,p,y3) double precision tk,p,y3 double precision rg,mw,a1,a2,a3 c** Cp of CH4: Fitted to the data in J. Chem. Eng. Data 8, 547 (1963) at T= 160-340 K rg=8.3145 mw=16.04 a1=-3.7677d-3 a2= 9.1107d-6 a3= 1.0178d-9 fdcp4dt=rg*(a1+2.d0*a2*tk+3.d0*a3*tk**2)/mw c fdcp4dt=0.0 return end
*************** Pressure derivative of Cp *********************
real*8 function fdcp2dp(tk,p,y1) double precision tk,p,y1 fdcp2dp=0.d0 return end
real*8 function fdcp3dp(tk,p,y2) double precision tk,p,y2 fdcp3dp=0.d0
236 return end
real*8 function fdcp4dp(tk,p,y3) double precision tk,p,y3 fdcp4dp=0.d0 return end
***************** Functions for Cpl ********************************** c** Cpl of Nonane real*8 function fcpl2(tk) double precision tk fcpl2=1.1061d-5*(tk**2)-2.8403d-3*(tk)+2.07130d0 return end c** Cpl of D2O real*8 function fcpl3(tk) double precision tk fcpl3=4.205d0 return end
******** Temperature derivative of gamma of gas mixture ***
real*8 function dgdt(tk,p,y1,y2,y3,w20,w30,w40,wmav0) double precision tk,p,y1,y2,y3,w20,w30,w40,wmav0 double precision rg,cpN,cp,gamma,dcpdt
rg=8.3145 cpN=1.0397
cp= (1.d0-w20-w30-w40)*cpN+w40*fcp4(tk,p,y3) & +w20*fcp2(tk,p,y1)+w30*fcp3(tk,p,y2) gamma=cp/(cp-rg/wmav0)
dcpdt= w20*fdcp2dt(tk,p,y1)+w30*fdcp3dt(tk,p,y2) & +w40*fdcp4dt(tk,p,y3)
dgdt=gamma*(1.d0-gamma)/cp*dcpdt
return end
237 ******** Pressure derivative of gamma of gas mixture ******
real*8 function dgdp(tk,p,y1,y2,y3,w20,w30,w40,wmav0) double precision tk,p,y1,y2,y3,w20,w30,w40,wmav0 double precision rg,cpN,cp,gamma,dcpdp
rg=8.3145 cpN=1.0397
cp= (1.d0-w20-w30-w40)*cpN+w40*fcp4(tk,p,y3) & +w20*fcp2(tk,p,y1)+w30*fcp3(tk,p,y2) gamma=cp/(cp-rg/wmav0)
dcpdp= w20*fdcp2dp(tk,p,y1)+w30*fdcp3dp(tk,p,y2) & +w40*fdcp4dp(tk,p,y3)
dgdp=gamma*(1.d0-gamma)/cp*dcpdp
return end
238 A.3: FORTRAN code used to calculate the experimental parameters using p and gfit of nonane and D2O from FTIR as input ccc this program version includes the ability to update the latent heat as a c function of temperature for h2o and d2o based on clausius- clapeyron c approximations to liquid-vapor equations. !chhfeb2001 ccc this version of the program calculates a "wet" isentrope based on the c measured dry isentrope and corrected for the differences in gamma. it c also starts the wet condensing flow integration on the desired data c point rather than on the wet isentrope to avoid any extraneous extra c shifts. ccc smoothes all of the good density data first, then integrates from an c initial value using finer integration grid (up to 5x) ccc modified to take in pressure data instead of density data ....jul97, jlc ccc 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 ccc this version has been modified for Nozzle H on train B with Velmex (PP) ccc RTD probe is calibrated and temperature calibration factor is included ccc Now nu.dat has "tempcal" and this program reads in the value and does c temperature calibration as "to(i)=to(i)+tempcal"...... jun02, PP ccc fc=g *wi/(w10+w20) was replaced by fc=g/(w10+w20) March04 Shinobu ccc Inert gas is a mixture of N2 and CH4 March04 Shinobu ccc tisd=pp0d(i)**c0*t0 was replaced by pp0d(i)**c3*t0 July05 Shinobu ccc Function fk has been corrected 3/31/2007 Shinobu, Hartawan ccc See Vol.6, p9 and Vol. 9, p68 ccc tempcal is not used from cal07. 10/17/2007 Shinobu ccc Gas constant was set to 8.3145. 10/18/2007 Shinobu ccc Changing for Nonane-D2O, new Properties for Nonane 17/06/2009 cc Cpliquid for nonane is changed for T dependence September 2010
implicit real*8(a-h,o-z) 239 real*8 fcon real*8 msq,msqw,mssq real*8 rg, pi, avog real*8 dotm,dotncal,pc10,pc20,zc10 real*8 p0, t0, tempcal real*8 xstart, xthroat real*8 tt0(1000),fc(1000),g(1000),u(1000), *rr0(1000),pp0(1000),pp0d(1000) real*8 tt0_is(1000),t_is(1000),rr0_is(1000),pp0_is(1000) real*8 t_is_s(1000),pp0_is_s(1000) real*8 aratio(1000),wg(1000),t(1000),tisd(1000) real*8 xd(1000),xw(1000),x(1000) real*8 dry(1000),dryf(1000),sdry(1000) real*8 wet(1000),swet(1000),wetf(1000) real*8 po(200),p(200),deltapo(200),deltap(200),to(200) real*8 deltadry(1000),deltadryf(1000) real*8 deltawet(1000),deltawetf(1000),dtemp(1000,20) real*8 m_1,mssq_is,m_0,m_2
real*8 mdry(1000) real*8 t_is_up(1000) c ccccccccccccccccccccccccccccccc ! Shinobu ccccccccc real*8 mssq_TDL,cp_TDL,cpr_TDL real*8 fc_TDL(1000),tt0_TDL(1000),u_TDL(1000),rr0_TDL(1000) real*8 g1_TDL(1000),g1_TDL2(1000),g1_TDL3(1000) real*8 g2_TDL(1000),g2_TDL2(1000),g2_TDL3(1000) real*8 x_TDL(1000),t_TDL(1000),ar_TDL(1000) real*8 cpv_TDL,cpc_TDL,y1_TDL,y2_TDL,y3_TDL cccccccccccccccccccccccccccccccccccccccccccccccccccccccccc character*30 dryfil,wetfil,a character*8 specie(2) character*60 progname c character*4 title(3,2) common /xval/ xs(1000)
*------nomenclature c dhc,fdhc(zc10,t(i)) latent heat of condensible vapor c pc10,pc20 condensible vapor pressure (read in 2*Torr, works in dyne/cm^2) c t(i) Temperature of inert in Kelvin c zc10 Initial molar fraction of condensible vapor1 (zc10+zc20=1) *------nomenclature
progname='nuetodd2o_irCH4MFC_DryCp2Up_cal07'
open(5,file='nu_CH4_MFC.dat',status='old') ! March04 Shinobu open(10,file='4pp.out',status='unknown') open(11,file='wilson.out',status='unknown') open(9,file='new4pp.out',status='unknown')
240 open(13,file='dtemp.out',status='unknown') c open(14,file='legend3.bat',status='unknown') c open(15,file='legend4.bat',status='unknown') open(7,file='upstream.out',status='unknown')
call echo
pi=3.14159d0 rg=8.3145d7 avog=6.022d23 c read two condensible species 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,*)tempcal !PP02 !RTD probe calibration added write(*,*)'tempcal = ',tempcal,' (not used)' c convert pressures to dyn/cm**2 pconv = 760.d0/1.01325d6 c read molecular weights of carrier (1), condensible (2,3) and CH4 (4) read(5,*)wmN,wm2,wm3,wm4 ! March04 Shinobu c read specific heats of gases read(5,*)cpN,cp2,cp3,cp4 ! March04 Shinobu c read latent heat, and specific heat of condensate read(5,*)dhc2,dhc3,cpc2,cpc3 c read starting value and the number of points in the output read(5,*) xstart2, ilast2 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. c istart >= 2, ifin < ilast, ni=1 (for useless roop, do k=1,ni) read(5,*)istart2, ifin2, 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 c read x values and all of the dry data p0, p(x), and the associated standard deviations. c correct the pressures i.e. baratron calibrations and pressure loss due to mesh. open(unit=4,file=dryfil,status='old')
241 c read total number of values in dry pressure data file read(4,1)a read(4,*)idend
dotncal=0.d0 p0dry=0.d0 c t0dry=273.15d0 +tempcal t0dry=0.0d0
do i=1,idend read(4,*)xd(i),po(i),deltapo(i),p(i),deltap(i),to(i),dummy2, * dummy3,dummy4,flowmain,dummy4,flowsub ! xd(i) in 0.01mm, po in 2*torr c po(i)=(po(i)*0.49967 + 2.19)-poloss !Shinobu cal04 c po(i)=po(i)/1.9888 !Shinobu cal05 c po(i)=po(i)*0.5025d0+0.457d0 !Shinobu cal07 c to(i)=to(i)*0.99147d0+0.510d0 ! RuiYang and Shinobu cal07 to(i)=to(i)*0.99375d0+0.352d0 ! Shinobu cal08 Vol.11, p.33 ccc Following equation includes the effect of poloss ccc po(i)=po(i)*0.5028d0+0.732d0 ! Shinobu cal08 Vol.11, p.54 c 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 dotncal=dotncal+(flowmain+flowsub)/idend/22.41d0 p0dry=p0dry+po(i) t0dry=t0dry+to(i) enddo close(unit=4) p0dry=p0dry/idend t0dry=t0dry/idend+273.15d0 c read the number of wet data sets and flow rate of CH4 read(5,*)ndata, dotCH4
wm0dry=(dotncal*wmN+dotCH4*wm4)/(dotncal+dotCH4) ! March04 Shinobu w40dry=dotCH4*wm4/(dotncal*wmN+dotCH4*wm4) y30dry=dotCH4/(dotncal+dotCH4) cp0dry=( dotncal*wmN*cpN+dotCH4*wm4*fcp4(t0dry,p0dry,y30dry) ) * / (dotncal*wmN+dotCH4*wm4) c write(*,*)'wm0dry,w40dry,y30dry,cp0dry=', c & wm0dry,w40dry,y30dry,cp0dry write(11,1302)dryfil
do kd = 1,ndata
242 read(5,*)ntype,entry1,entry2,entry3 ! March04 Shinobu if(ntype.eq.0)then !pressure input (torr) pc10=entry1 pc20=entry2 tCH4=entry3 ! March04 Shinobu else if(ntype.eq.1)then !massflow and weight fraction input dotm=entry1 wfc10=entry2 wfc20=1.0d0-wfc10 c write(7,*)wfc10 tCH4=entry3 ! March04 Shinobu else write(*,*)'need to specify pressure (0)' write(*,*)'or mass flow with first weight fraction input(1)' stop end if 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. tN=0.d0 t0set=0.d0 do i=1,idenw read(4,*)xw(i),po(i),deltapo(i),p(i),deltap(i),to(i),dummy1, * dummy2,dummy3,flowmain,dummy4,flowsub !po in 2*torr c po(i)=po(i)/1.9888 !Shinobu cal05 c po(i)=po(i)*0.5025+0.457 !Shinobu cal07 ccc Following equation includes the effect of poloss ccc po(i)=po(i)*0.5028d0+0.732d0 ! Shinobu cal08 Vol.11, p.54 c t0set=t0set+to(i)/idenw c to(i)=to(i)+tempcal !ppaci02! RTD probe calibration c to(i)=to(i)*0.99147d0+0.510d0 ! RuiYang and Shinobu cal07 to(i)=to(i)*0.99375d0+0.352d0 ! Shinobu cal08 Vol.11, p.33 wet(i)=p(i)/po(i) deltawet(i)=wet(i) * *((deltap(i)/p(i))**2.0+(deltapo(i)/po(i))**2.0)**0.5
243 c write(*,*) xw(i), wet(i), deltawet(i) !debug tN=tN+(flowmain+flowsub)/idenw/22.41d0 enddo cccccccccccccccccccccccccccccc ! Shinobu cccc read(4,*) iden_TDL write(9,*)'number of lines of g is ',iden_TDL do i=1,iden_TDL read(4,*) x_TDL(i),g1_TDL2(i),g2_TDL2(i) enddo read(4,*) g1inf read(4,*) g2inf write(9,*)'g1inf is ',g1inf write(9,*)'g2inf is ',g2inf ccccccccccccccccccccccccccccccccccccccccccccc
close(unit=4) c figure out the average stagnant pressure and temperature
p0=0.0 t0=0.0 do i=1,idenw p0=p0+po(i)
t0=t0+to(i) enddo p0=p0/idenw t0=t0/idenw+273.15 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'
allflux=tN+tCH4+dotm*wfc10/wm2+dotm*wfc20/wm3 wm1=(tN*wmN+tCH4*wm4)/(tN+tCH4) w40=tCH4*wm4/(tN*wmN+tCH4*wm4+dotm) y30=tCH4/allflux c !chh99 c now figure out pcondensible from calibration and average properties if(ntype.eq.1)then !now calculate pcondensible pc10=p0*dotm*wfc10/wm2/allflux ! June05 Shinobu
244 write(*,*)'pc10= ',pc10,' torr' pc20=p0*dotm*wfc20/wm3/allflux write(*,*)'pc20= ',pc20,' torr' endif c convert pressures to dyn/cm**2 p0=p0/pconv pct0=pc10+pc20 pc10=pc10/pconv pc20=pc20/pconv if((pc10+pc20).lt.1.d-18) then zc10=0.d0 else zc10=pc10/(pc10+pc20) !chh22.02.01 endif y10=pc10/p0 y20=pc20/p0 c calculate stagnation gas mass density and condensible monomer mass c density (g/cm**3) c w2,w3 are mass fraction of condensible vapor in gas wmav=(wm1*(p0-pc10-pc20)+wm2*pc10+wm3*pc20)/p0 w20=wm2*pc10/p0/wmav w30=wm3*pc20/p0/wmav wi=1.d0-w20-w30 wN0=wi-w40 c gw17-2-00 assuming vapor condenses at constant composition let's define c a fictitious mean condensible vapor molecular weight wmc if((pc10+pc20).lt.1.d-18) then wmc=0.d0 else wmc=(wm2*pc10+wm3*pc20)/(pc10+pc20) endif c also let's save the inital average molecular weight wmav0=wmav
cp0= wN0*cpN+w40*fcp4(t0,p0,y30) & +w20*fcp2(t0,p0,y10)+w30*fcp3(t0,p0,y20)
gamma=cp0dry/(cp0dry-rg*1.d-7/wm0dry) !n2 gamma gamma0=cp0/(cp0-rg*1.d-7/wmav) !initial mixture gamma rhog0=p0/rg/t0*wmav write(*,*)'wmav',' w20',' w30',' wi',' cp0',' gamma0' !chh061098 write(*,*)wmav, w20,w30,wi,cp0,gamma0 c calculate various exponents and constants involving gamma
245 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) c2 = (gamma0+1.d0)/2.d0 c0 = (gamma0-1.d0)/gamma0 c3 = (gamma-1.d0)/gamma 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 tstt0=pstp0**c3
*********** Values at throat under Dry condition, Shinobu *************
stepm=250.0 nstepm=250 dm_1=1.d0/stepm m_1=0.d0+dm_1
t_0=t0dry p_0=p0dry
t_1=t_0 p_1=p_0 g_1=gamma cp_1=cp0dry
do i=2,nstepm
fk_1=( dgdt(t_1,p_1,0.d0,0.d0,y30dry,0.d0,0.d0,w40dry,wm0dry) & *t_1 + p_1*g_1/(g_1-1.d0)* & dgdp(t_1,p_1,0.d0,0.d0,y30dry,0.d0,0.d0,w40dry,wm0dry) )/ & ( 2.d0+(g_1-1.d0)*m_1*m_1 ) dg_1=-2.d0*fk_1*m_1*(g_1-1.d0)*2.d0*dm_1/ & (1.d0+fk_1*m_1*m_1*(g_1-1.d0)/g_1) dt_1=-t_1/(2.d0+(g_1-1.d0)*m_1*m_1)* & ( 2.d0*(g_1-1.d0)*m_1*2.d0*dm_1 + & m_1*m_1*(g_1-1.d0)/g_1*dg_1 ) dp_1=p_1/t_1*g_1/(g_1-1.d0)*dt_1
246 dr_1=r_1/t_1/(g_1-1.d0)*dt_1
t_2=t_0+dt_1 p_2=p_0+dp_1 r_2=r_0+dr_1 cp_2=(1.d0-w40dry)*cpN+w40dry*fcp4(t_2,p_2,y30dry) g_2=cp_2/(cp_2-rg*1.d-7/wm0dry)
t_0=t_1 p_0=p_1 t_1=t_2 p_1=p_2 cp_1=cp_2 g_1=g_2 m_1=m_1+dm_1 c write(*,*)'m_1,cp_1,g_1',m_1,cp_1,g_1 c write(*,*)'dg_1, dt_1, dp_1, ',dg_1,dt_1,dp_1 c write(*,*)'p_1/p0, t_1/t0',p_1/p0,t_1/t0
enddo
tstcpdry=t_1 pstcpdry=p_1
write(*,*) ' pstp0, tstt0 (for constant Cp) =', pstp0,tstt0 write(*,*) 'pstcpdry/p0dry, tstcpdry/t0dry = ', & pstcpdry/p0dry, tstcpdry/t0dry ******************************************************************** ***
pstp0dry=pstcpdry/p0dry do i=1,idend c write(*,*) i, xd(i), dry(i) !debug if((dry(i).gt. pstp0dry).and.(dry(i+1).le. pstp0dry))then c write(*,*) 'true' !debug xthroat=( pstp0dry-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 do i=1,idend x(i)=(xd(i)-xthroat)/1000.0 ! in units of cm !Shinobu for Velmex on Train B enddo c now do linear interpolation to get fixed x intervals
247 ************************************************ ixstart=1 xstart= int(x(1)*10.d0)/10.d0 ilast=ilast2+int( (xstart2-xstart)/0.1+0.1 ) write(*,*) 'xstart= ',xstart ************************************************ 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 enddo
write(*,*) 'can not interpolate for point', j
5 continue enddo c we now have an array dryf(j) at fixed xs(j) intervals. now put 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
248 1201 format(f10.4,f10.4,g14.4,f10.4,f10.4,g14.4) 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
*********** Values at throat under Wet condition, Shinobu *************
stepm=250.0 nstepm=250 dm_1=1.d0/stepm m_1=0.d0+dm_1
t_0=t0 p_0=p0 r_0=rhog0
t_1=t_0 p_1=p_0 r_1=r_0 g_1=gamma0 cp_1=cp0
do i=2,nstepm
fk_1=( dgdt(t_1,p_1,y10,y20,y30,w20,w30,w40,wmav0) & *t_1 + p_1*g_1/(g_1-1.d0)* & dgdp(t_1,p_1,y10,y20,y30,w20,w30,w40,wmav0) )/ & ( 2.d0+(g_1-1.d0)*m_1*m_1 ) dg_1=-2.d0*fk_1*m_1*(g_1-1.d0)*2.d0*dm_1/ & (1.d0+fk_1*m_1*m_1*(g_1-1.d0)/g_1) dt_1=-t_1/(2.d0+(g_1-1.d0)*m_1*m_1)* & ( 2.d0*(g_1-1.d0)*m_1*2.d0*dm_1 + & m_1*m_1*(g_1-1.d0)/g_1*dg_1 ) dp_1=p_1/t_1*g_1/(g_1-1.d0)*dt_1 dr_1=r_1/t_1/(g_1-1.d0)*dt_1
t_2=t_0+dt_1 p_2=p_0+dp_1 r_2=r_0+dr_1 cp_2= wN0*cpN+w40*fcp4(t_2,p_2,y30) & +w20*fcp2(t_2,p_2,y10)+w30*fcp3(t_2,p_2,y20) g_2=cp_2/(cp_2-rg*1.d-7/wmav)
t_0=t_1 p_0=p_1 r_0=r_1
249 t_1=t_2 p_1=p_2 r_1=r_2 cp_1=cp_2 g_1=g_2 m_1=m_1+dm_1 c write(*,*)'m_1,cp_1,g_1',m_1,cp_1,g_1 c write(*,*)'dg_1, dt_1, dp_1, ',dg_1,dt_1,dp_1 c write(*,*)'p_1/p0, t_1/t0',p_1/p0,t_1/t0
enddo
gammam=g_1 tstarcp=t_1 pstarcp=p_1 rstarcp=r_1 ustarcp=dsqrt(gammam*rg*tstarcp/wmav)
write(*,*)'p*/p0, t*/t0, r*/r0 (constant Cp) ', & pstp0w,1.d0/c2,c2**erho write(*,*)'p_1/p0, t_1/t0, r_1/r0 ',p_1/p0,t_1/t0,r_1/rhog0 write(*,*) 'gamma0, gammam',gamma0,gammam ******************************************************************** ***
do i=1,idenw c write(*,*) i, xw(i), wet(i) !debug if((wet(i).gt.pstarcp/p0).and.(wet(i+1).le.pstarcp/p0))then c write(*,*) 'true' !debug xthroat=xw(i)+(pstarcp/p0-wet(i))/ & (wet(i+1)-wet(i))*(xw(i+1)-xw(i)) go to 5002 endif enddo 5002 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. !chh110698 ixstart=0 !chh110698 do i=1,idenw x(i)=(xw(i)-xthroat)/1000.0 ! in units of cm if(x(i).le.xstart)ixstart=i !chh110698 enddo c write(*,*) 'ixstart= ',ixstart !chh110698 write(*,*) 'throat shifted' !debug
250 c now do linear interpolation to get fixed x intervals lasti=ixstart !chh110698 do j=1,ilast c xs values have already been assigned in dry data analysis c write(*,*) xs(j) !debug do i=lasti,idenw !chh110698 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 !chh110698 goto 6 endif enddo
write(*,*) 'can not interpolate for point', j 6 continue enddo cccccccccccccccccccccccccccccccccc ! Shinobu lasti=1 do j=1,ilast c write(*,*) xs(j) !debug do i=lasti,iden_TDL if((x_TDL(i).le.xs(j)).and.(x_TDL(i+1).gt.xs(j))) then g1_TDL3(j)=g1_TDL2(i)+(xs(j)-x_TDL(i))* & (g1_TDL2(i+1)-g1_TDL2(i))/(x_TDL(i+1)-x_TDL(i)) g2_TDL3(j)=g2_TDL2(i)+(xs(j)-x_TDL(i))* & (g2_TDL2(i+1)-g2_TDL2(i))/(x_TDL(i+1)-x_TDL(i))
lasti=i !chh110698 goto 62 endif enddo
write(*,*) 'can not interpolate for point', j 62 continue enddo ccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccc
c we now have an array wetf(j) at fixed xs(j) intervals. now put c through smoothing routine. write(*,*) 'put through smoothing' c c smooth wet pressure values
251 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 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'
********************************************** ifin=ifin2+int( (xstart2-xstart)/0.1+0.1 ) istart0=1 **********************************************
npts=ifin-istart0+1 nnpts=(npts-1)*nint+1 jinit=nnpts+2*nint+istart0-1 do i=ifin+1,istart0,-1 delx=xs(i)-xs(i-1) delprd = sdry(i)-sdry(i-1) c delprwi = sweti(i)-sweti(i-1) delprw = swet(i)-swet(i-1) delg1=g1_TDL3(i)-g1_TDL3(i-1) ! Shinobu delg2=g2_TDL3(i)-g2_TDL3(i-1) ! Shinobu
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 if(dabs(xs(j)).LT.1.d-4) ithroat=j
252 pp0d(j) = sdry(i)-delprd*fint c pp0i(j) = sweti(i)-delprwi*fint pp0(j) = swet(i)-delprw *fint g1_TDL(j)=g1_TDL3(i)-delg1*fint ! Shinobu g2_TDL(j)=g2_TDL3(i)-delg2*fint ! Shinobu jp=jp+1 enddo enddo ifin1=istart0+nnpts-1
***************************************************************** istart= istart0+ & int( (xstart2-xstart)/0.1+istart2-istart0+0.1 )*nint write(9,*)'ithroat =',ithroat *****************************************************************
*** Pressure and temperature upstream of the integration region **** t_0=tstarcp p_0=pstarcp
cp_0= wN0*cpN+w40*fcp4(t_0,p_0,y30) & +w20*fcp2(t_0,p_0,y10)+w30*fcp3(t_0,p_0,y20) g_0=cp_0/(cp_0-rg*1.d-7/wmav)
dp=( pp0(ithroat+1)-pp0(ithroat-1) )*p0 dt=t_0/p_0*(g_0-1.d0)/g_0*dp
t_is_up(ithroat)=tstarcp t_is_up(ithroat+1)=tstarcp+dt/2.d0 t_is_up(ithroat-1)=tstarcp-dt/2.d0
t_1=t_is_up(ithroat+1) p_1=pp0(ithroat+1)*p0
do i=ithroat+2,istart cp_1= wN0*cpN+w40*fcp4(t_1,p_1,y30) & +w20*fcp2(t_1,p_1,y10)+w30*fcp3(t_1,p_1,y20) g_1=cp_1/(cp_1-rg*1.d-7/wmav)
dp=( pp0(i)-pp0(i-2) )*p0 dt=t_1/p_1*(g_1-1.d0)/g_1*dp
t_is_up(i)=t_is_up(i-2)+dt
t_1=t_is_up(i) p_1=pp0(i)*p0 enddo
t_1=t_is_up(ithroat-1) p_1=pp0(ithroat-1)*p0
253 do i=ithroat-2,istart0,-1 cp_1= wN0*cpN+w40*fcp4(t_1,p_1,y30) & +w20*fcp2(t_1,p_1,y10)+w30*fcp3(t_1,p_1,y20) g_1=cp_1/(cp_1-rg*1.d-7/wmav)
dp=( pp0(i)-pp0(i+2) )*p0 dt=t_1/p_1*(g_1-1.d0)/g_1*dp
t_is_up(i)=t_is_up(i+2)+dt
t_1=t_is_up(i) p_1=pp0(i)*p0 enddo
write(7,700) 700 format(' x(cm) pp0 p(Torr) Tisw(K)')
do i=istart0,istart write(7,710) xs(i),pp0(i),pp0(i)*p0*pconv,t_is_up(i) 710 format(f7.2,f8.4,2f10.2) enddo close(unit=7) ******************************************************************** ***
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
***********Values at the start point of integration for Dry, Shinobu **********
stepp=100.0 nstepp=100 dp_1=(pp0d(istart-1)*p0dry-pstcpdry)/stepp t_0=tstcpdry p_0=pstcpdry cp_0=(1.d0-w40dry)*cpN+w40dry*fcp4(t_0,p_0,y30dry) g_0=cp_0/(cp_0-rg*1.d-7/wm0dry) m_0=1.d0 fk_0=( dgdt(t_0,p_0,0.d0,0.d0,y30dry,0.d0,0.d0,w40dry,wm0dry) & *t_0 + p_0*g_0/(g_0-1.d0)* & dgdp(t_0,p_0,0.d0,0.d0,y30dry,0.d0,0.d0,w40dry,wm0dry) )/ & ( 2.d0+(g_0-1.d0)*m_0*m_0 ) a_0=1.d0
254 dt_0=t_0/p_0*(g_0-1.d0)/g_0*dp_1 dm_0= -(2.d0+(g_0-1.d0)*m_0**2) & *( g_0+fk_0*(m_0**2)*(g_0-1.d0) ) & /(2.d0*(g_0**2)*m_0)/p_0*dp_1 da_0= -a_0*(m_0**2-1.d0)/(g_0*m_0**2)/p_0*dp_1
t_1=t_0+dt_0 m_1=m_0+dm_0 p_1=p_0+dp_1 a_1=a_0+da_0 cp_1=(1.d0-w40dry)*cpN+w40dry*fcp4(t_1,p_1,y30dry) g_1=cp_1/(cp_1-rg*1.d-7/wm0dry) fk_1=( dgdt(t_1,p_1,0.d0,0.d0,y30dry,0.d0,0.d0,w40dry,wm0dry) & *t_1 + p_1*g_1/(g_1-1.d0)* & dgdp(t_1,p_1,0.d0,0.d0,y30dry,0.d0,0.d0,w40dry,wm0dry) )/ & ( 2.d0+(g_1-1.d0)*m_1*m_1 )
do i=2, nstepp dt_1=t_1/p_1*(g_1-1.d0)/g_1*2.d0*dp_1 dm_1= -(2.d0+(g_1-1.d0)*m_1**2) & *( g_1+fk_1*(m_1**2)*(g_1-1.d0) ) & /(2.d0*(g_1**2)*m_1)/p_1*2.d0*dp_1 da_1= -a_1*(m_1**2-1.d0)/(g_1*m_1**2)/p_1*2.d0*dp_1
t_2=t_0+dt_1 m_2=m_0+dm_1 a_2=a_0+da_1 p_2=p_0+2.d0*dp_1
cp_2=(1.d0-w40dry)*cpN+w40dry*fcp4(t_2,p_2,y30dry) g_2=cp_2/(cp_2-rg*1.d-7/wm0dry) fk_2=( dgdt(t_2,p_2,0.d0,0.d0,y30dry,0.d0,0.d0,w40dry,wm0dry) & *t_2 + p_2*g_2/(g_2-1.d0)* & dgdp(t_2,p_2,0.d0,0.d0,y30dry,0.d0,0.d0,w40dry,wm0dry) )/ & ( 2.d0+(g_2-1.d0)*m_2*m_2 )
t_0=t_1 m_0=m_1 a_0=a_1 p_0=p_1 t_1=t_2 m_1=m_2 a_1=a_2 g_1=g_2 fk_1=fk_2 p_1=p_2
255 c write(*,*)'dt_1,dm_1,da_1 =',dt_1,dm_1,da_1
enddo
tisd(istart-1)=t_1 aratio(istart-1)=a_1 ar_TDL(istart-1)=a_1 mdry(istart-1)=m_1
dp_2=dp_1+(pp0d(istart)-pp0d(istart-1))*p0dry dt_1=t_1/p_1*(g_1-1.d0)/g_1*dp_2 dm_1= -(2.d0+(g_1-1.d0)*m_1**2) & *( g_1+fk_1*(m_1**2)*(g_1-1.d0) ) & /(2.d0*(g_1**2)*m_1)/p_1*dp_2 da_1= -a_1*(m_1**2-1.d0)/(g_1*m_1**2)/p_1*dp_2
t_2=t_0+dt_1 m_2=m_0+dm_1 a_2=a_0+da_1 p_2=p_0+dp_2
tisd(istart)=t_2 mdry(istart)=m_2 aratio(istart)=a_2 ar_TDL(istart)=a_2 ******************************************************************** ************* 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. c msqw = c10*((1.d0/pp0(istart-1))**c0-1.d0) c tt0(istart-1)=1.d0/(1.d0+msqw/c10) c rr0(istart-1)=(1.d0+msqw/c10)**erho c msqw = c10*((1.d0/pp0( istart))**c0-1.d0) c tt0(istart)=1.d0/(1.d0+msqw/c10) c rr0(istart)=(1.d0+msqw/c10)**erho
***********Values at the start point of integration for Wet, Shinobu **********
stepp=100.0 nstepp=100 dp_1=(pp0(istart-1)*p0-pstarcp)/stepp
t_0=tstarcp p_0=pstarcp r_0=rstarcp
256 g_0=gammam
dt_0=t_0/p_0*(g_0-1.d0)/g_0*dp_1 dr_0=r_0/p_0/g_0*dp_1 t_1=t_0+dt_0 r_1=r_0+dr_0 p_1=p_0+dp_1 cp_1= wN0*cpN+w40*fcp4(t_1,p_1,y30) & +w20*fcp2(t_1,p_1,y10)+w30*fcp3(t_1,p_1,y20) g_1=cp_1/(cp_1-rg*1.d-7/wmav)
do i=2,nstepp
dt_1=t_1/p_1*(g_1-1.d0)/g_1*2.d0*dp_1 dr_1=r_1/p_1/g_1*2.d0*dp_1 t_2=t_0+dt_1 r_2=r_0+dr_1 p_2=p_0+2.d0*dp_1 cp_2= wN0*cpN+w40*fcp4(t_2,p_2,y30) & +w20*fcp2(t_2,p_2,y10)+w30*fcp3(t_2,p_2,y20) g_2=cp_2/(cp_2-rg*1.d-7/wmav)
t_0=t_1 r_0=r_1 p_0=p_1 t_1=t_2 r_1=r_2 cp_1=cp_2 g_1=g_2 p_1=p_2
enddo
tt0_is(istart-1)=t_1/t0 rr0_is(istart-1)=r_1/rhog0 pp0_is(istart-1)=p_1/p0
msqw = c10*((1.d0/pp0(istart-1))**c0-1.d0) write(*,*)'pp0(istart-1), tt0, rr0', & pp0(istart-1),1.d0/(1.d0+msqw/c10),(1.d0+msqw/c10)**erho write(*,*)'p_1/p0,t_1/t0,r_1/rhog0', & pp0_is(istart-1),tt0_is(istart-1),rr0_is(istart-1)
dp_2=dp_1+(pp0(istart)-pp0(istart-1))*p0 dt_1=t_1/p_1*(g_1-1.d0)/g_1*dp_2 dr_1=r_1/p_1/g_1*dp_2 t_2=t_0+dt_1 r_2=r_0+dr_1
tt0_is(istart)=t_2/t0 rr0_is(istart)=r_2/rhog0
257 pp0_is(istart)=(p_0+dp_2)/p0
msqw = c10*((1.d0/pp0( istart))**c0-1.d0) write(*,*)'pp0(istart), tt0, rr0 ', & pp0(istart),1.d0/(1.d0+msqw/c10),(1.d0+msqw/c10)**erho write(*,*)'p_2/p0,t_2/t0,r_2/rhog0', & pp0_is(istart),tt0_is(istart),rr0_is(istart)
tt0(istart-1)=tt0_is(istart-1) tt0_TDL(istart-1)=tt0_is(istart-1) rr0(istart-1)=rr0_is(istart-1) rr0_TDL(istart-1)=rr0_is(istart-1) tt0(istart)=tt0_is(istart) tt0_TDL(istart)=tt0_is(istart) rr0(istart)=rr0_is(istart) rr0_TDL(istart)=rr0_is(istart) ******************************************************************** ***
g(istart)=0.d0 g(istart-1)=0.d0 fc(istart)=0.0d0 !fraction condensed fc_TDL(istart)=0.0d0
write(10,1024) progname 1024 format('Program: ',a60)
write(10,1011)p0*pconv,devp0,t0-273.15, t0set, rhog0*1.0d3 !4pp plots write(10,1010) dotm,specie(1),wfc10 !4pp plots 1010 format('Weight flux of condensable =',f6.2, & ' g/min , Fraction of ',a, '=',f7.3) !4pp plots 1011 format('p0= ',f6.2,'+/-',f4.2,' Torr T0=',f6.2, & ' C (set T0=',f6.2,') rho0=', e11.4,' kg/m3') !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) allflux, (dotncal+dotCH4) !4pp plots 1013 format( 'Total mole flux =', & f6.3,' mol/min (',f6.3,' in Dry )' ) !4pp plots write(10,1019) tCH4, dotCH4 !4pp plots 1019 format( ' CH4 mole flux =', & f6.3,' mol/min (',f6.3,' in Dry )' ) !4pp plots write(10,1014)wetfil,dryfil !4pp plots
258 1014 format('@subtitle \"',a30,'with dry trace ',a30,'\"') !4pp plots write(10,1015) !4pp plots 1015 format(' x(cm) u(m/s) T(K) p/p0 Tis p/p0_is', &' MoleFract. g g/g_inf A/A* r/r0', &' Tisd p/p0_isd') !4pp plots c write(12,1016)kd,pc10*pconv,specie(1),pc20*pconv, c +specie(2) !4pp plots c 1016 format('legend string ',i2,' \"',2(f7.4,'torr ',a),'\"') !4pp plots c write(14,1017)kd, wetfil !4pp plots c 1017 format('legend string ',i2,' \"',a13,'\"') !4pp plots c write(15,1018)kd-1,p0*pconv,devp0,t0-273.15 !4pp plots c 1018 format('legend string ',i2,' \"',f6.2,'+/-',f4.2,'torr ', c & f6.2,'celsius"') !4pp plots write(9,1037) !4pp plots in new output file 1037 format(' x(cm) u_PTM T_PTM g_PTM T_is Tptm-Tis P/po_is', &' P/po g/g_inf T_tdl A/A*_PTM A/A*_TDL g1/g1inf_TDL', &' g2/g2inf_TDL r/ro_TDL u_TDL')
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
*********** Integration of the isentropic curve for Dry, Shinobu **************
dp_dry=( pp0d(i+1)-pp0d(i-1) )*p0dry/2.d0
p_dry=pp0d(i)*p0dry cp_dry=(1.d0-w40dry)*cpN + w40dry*fcp4(tisd(i),p_dry,y30dry) g_dry=cp_dry/(cp_dry-rg*1.d-7/wm0dry) fk_dry=( dgdt(tisd(i),p_dry,0.d0,0.d0,y30dry, & 0.d0,0.d0,w40dry,wm0dry) & *tisd(i) + p_dry*g_dry/(g_dry-1.d0)* & dgdp(tisd(i),p_dry,0.d0,0.d0,y30dry, & 0.d0,0.d0,w40dry,wm0dry) )/ & ( 2.d0+(g_dry-1.d0)*mdry(i)**2 )
dt_dry=tisd(i)/p_dry*(g_dry-1.d0)/g_dry*dp_dry dm_dry= -( 2.d0+(g_dry-1.d0)*mdry(i)**2 ) & *( g_dry+fk_dry*(mdry(i)**2)*(g_dry-1.d0) ) & /(2.d0*(g_dry**2)*mdry(i))/p_dry*dp_dry da_dry= -aratio(i)*(mdry(i)**2-1.d0)/(g_dry*mdry(i)**2)/
259 & p_dry*dp_dry
tisd(i+1)=tisd(i-1)+2.d0*dt_dry mdry(i+1)=mdry(i-1)+2.d0*dm_dry aratio(i+1)=aratio(i-1)+2.d0*da_dry
dar=dlog(aratio(i+1)/aratio(i-1))/2.d0 c write(*,*) 'Integration of the isentropic curve for Dry OK' ******************************************************************** ************* ccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccc c Arearatio for constant Cp of CH4 c c msq = c1*((1.d0/pp0d(i))**c3-1.d0) c aratio(i)= dsqrt(((c1/eam2*(1.d0+msq/c1))**eam2)/msq) c msq = c1*((1.d0/pp0d(i+1))**c3-1.d0) c aratio(i+1) = dsqrt(((c1/eam2*(1.d0+msq/c1))**eam2)/msq) c dar=dlog(aratio(i+1)/aratio(i-1))/2.d0 c c cccccccccccccccccccccccccccccccccccccccccccccccccccccccccccc
*********** Integration of the isentropic curve for Wet, Shinobu **************
dp=(pp0(i+1)-pp0(i-1))/2.d0
t_is(i)=tt0_is(i)*t0
u_is=ustarcp*rstarcp/rhog0/rr0_is(i)/aratio(i) mssq_is=(u_is/ustarcp)**2 cp_is= wN0*cpN+w40*fcp4(t_is(i),pp0_is(i)*p0,y30) & +w20*fcp2(t_is(i),pp0_is(i)*p0,y10) & +w30*fcp3(t_is(i),pp0_is(i)*p0,y20) cpr_is=cp_is/cp0 hpara_is=1.d0-(gamma0-1.d0)/gamma0/cpr_is
tempA=(1.d0-t_is(i)/tstarcp/gammam/mssq_is)/rr0_is(i) tempG=t0/tstarcp/gammam/mssq_is tempJ=rr0_is(i)*tt0_is(i)/ & (hpara_is-t_is(i)/tstarcp/gammam/mssq_is)
dtt0_is=(tt0_is(i)-tempA*tempJ)*dar dpp0_is=-tempJ*dar drr0_is=-(rr0_is(i)+tempG*tempJ)*dar
tt0_is(i+1)=tt0_is(i-1)+2.d0*dtt0_is pp0_is(i+1)=pp0_is(i-1)+2.d0*dpp0_is rr0_is(i+1)=rr0_is(i-1)+2.d0*drr0_is
260 c write(*,*) 'Integration of the isentropic curve for Wet OK' *************** Smoothing, Shinobu ************************************ t_is_s(i)=t0*( tt0_is(i-1)+2.d0*tt0_is(i)+tt0_is(i+1) )/4.d0 pp0_is_s(i)=( pp0_is(i-1)+2.d0*pp0_is(i)+pp0_is(i+1) )/4.d0 c t_is_s(i)=t0*tt0_is(i) c pp0_is_s(i)=pp0_is(i) ******************************************************************** *** c write(*,*)'mssq,mssq_is',mssq,mssq_is c write(*,*)'cp,cp_is',cp,cp_is c write(*,*)'hpara,hpara_is',hpara,hpara_is c write(*,*)'gamma0,gammam',gamma0,gammam c write(*,*)'tempA, tempJ,tempG',tempA,tempJ,tempG c write(*,*) rr0_is(i) c write(*,*)'dar',dar c write(*,*)'dtt0,dpp0',dtt0_is,dpp0_is c write(*,*)'mdry(i), Mach ,fk_dry=', c & mdry(i), dsqrt(c1*((1.d0/pp0d(i))**c3- 1.d0)),fk_dry
******************************************************************** *** t(i)=tt0(i)*t0 t_TDL(i)=tt0_TDL(i)*t0 u(i)=ustarcp*rstarcp/rhog0/rr0(i)/aratio(i) u_TDL(i)=ustarcp*rstarcp/rhog0/rr0_TDL(i)/ar_TDL(i) ! harshad mssq=(u(i)/ustarcp)**2 mssq_TDL=(u_TDL(i)/ustarcp)**2 c c fcon=dotm * (wfc10/wm2 + (1-wfc10)/wm3) c c if((pc10+pc20).lt.1.d-18) then c c y1=0.d0 c c y2=0.d0 c else c y1=(1-fc(i))*fcon/(allflux-fc(i)*fcon)*y10/(y10+y20) c y2=(1-fc(i))*fcon/(allflux-fc(i)*fcon)*y20/(y10+y20) c endif c y3=y30*allflux/(allflux-fc(i)*fcon) c write(*,*)'(y1+y2+y3+tN/allflux*y3/y30), (wN0+w20+w30+w40)', c & y1+y2+y3+tN/allflux*y3/y30, wN0+w20+w30+w40 c harshad-y values required only in case of clustering y1=0 y1_TDL=0 y2=0 y2_TDL=0 y3=0
261 y3_TDL=0 c gw2-17-00 update specific heat if((pc10+pc20).lt.1.d-18) then cpv=0.d0 cpc=0.d0 else cpv=( w20*fcp2(t(i),pp0(i)*p0,y1) + & w30*fcp3(t(i),pp0(i)*p0,y2) )/(w20+w30) cpc=( w20*fcpl2(t(i))+w30*fcpl3(t(i)) )/(w20+w30) endif cp= wN0*cpN+w40*fcp4(t(i),pp0(i)*p0,y3) & +(w20+w30-g(i))*cpv+g(i)*cpc cpr=cp/cp0 c gw2-17-00 update specific heat for TDL if((pc10+pc20).lt.1.d-18) then cpv_TDL=0.d0 cpc_TDL=0.d0 else cpv_TDL=(w20*(1- g1_TDL(i)/g1inf)*fcp2(t_TDL(i),pp0(i)*p0,y1_TDL) & + w30*(1- g2_TDL(i)/g2inf)*fcp3(t_TDL(i),pp0(i)*p0,y2_TDL)) & /(w20*(1-g1_TDL(i)/g1inf)+w30*(1-g2_TDL(i)/g2inf)) if((g1_TDL(i)).lt.1.d-18) then cpc_TDL=cpv_TDL else cpc_TDL=(w20*(g1_TDL(i)/g1inf)*fcpl2(t_TDL(i)) & + w30*(g2_TDL(i)/g2inf)*fcpl3(t_TDL(i)))/ & (w20*(g1_TDL(i)/g1inf)+w30*(g2_TDL(i)/g2inf)) endif endif c write(9,*)'cpvapor is ',cpv_TDL c write(9,*)'cpLIQ is ',cpc_TDL c write(9,*)'cpliq for nonane is ',fcpl2(t_TDL(i)) c write(9,*)'cpliq for d2o is ',fcpl3(t_TDL(i)) c write(9,*)'w2o is ',w20 c write(9,*)'w3o is ',w30 cp_TDL= wN0*cpN+w40*fcp4(t_TDL(i),pp0(i)*p0,y3_TDL) & +(w20+w30-g1_TDL(i)- g2_TDL(i))*cpv_TDL+ & (g1_TDL(i)+g2_TDL(i))*cpc_TDL c write(9,*)'Cp_TDL IS ',cp_TDL cpr_TDL=cp_TDL/cp0 c gw2-17-00 update "mu/(1-g)" = wmu, and related factors if((pc10+pc20).lt.1.d-18) then wmu=wm1 wg(i)=0.d0 else wmu=wm1*wmc/(wi*wmc+(w20+w30-g(i))*wm1)
262 wg(i)=wmu/(wmc) endif wmuu0=wmu/wmav0 c gw2-17-00 update "mu/(1-g)" = wmu_TDL, and related factors if((pc10+pc20).lt.1.d-18) then wmu_TDL=wm1 wg_TDL=0.d0 else wmc_TDL= (wm2*(1-g1_TDL(i)/g1inf)*pc10+ & wm3*(1-g2_TDL(i)/g2inf)*pc20)/( pc10*(1-g1_TDL(i)/g1inf)+ & pc20*(1-g2_TDL(i)/g2inf)) wmu_TDL=wm1*wmc_TDL/(wi*wmc_TDL+(w20+w30-g1_TDL(i)- & g2_TDL(i))*wm1) wg_TDL=wmu_TDL/(wmc_TDL) endif wmuu0_TDL=wmu_TDL/wmav0
hpara=wmuu0-(gamma0-1.d0)/gamma0/cpr dr=dp/tstarcp*t0/gammam/mssq-rr0(i)*dar dgp=(hpara-t(i)/tstarcp/mssq/gammam)/rr0(i)*dp+tt0(i)*dar if((pc10+pc20).lt.1.d-18) then dg=0.d0 else dg=dgp*cp*t0/(fdhc(wfc10,t(i))-cp*t(i)*wg(i)) ! Shinobu endif
c gw2-17-00 update dtt0 dtt0=(wmuu0-t(i)/tstarcp/gammam/mssq)/rr0(i)*dp+ & tt0(i)*(dar+wg(i)*dg)
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
if((w20+w30).gt.0.0)then fc(i+1)=g(i+1)/(w20+w30) ! March04 Shinobu else fc(i+1)=0.0 end if c c write(*,*) 'Integration of the Wet trace OK'
ap_TDL=wmuu0_TDL*gamma0-(gamma0-1.d0)/cpr_TDL ! Shinobu h_TDL=ap_TDL/gamma0 ! Shinobu
263 ccccccccccccccccccc ! Harshad cccccccccccccccccccccccc cccthis part takes dg1,dg2 as inputs,calculates new T/to,r/ro,u,a/a*cccc dg1_TDL=(g1_TDL(i+1)-g1_TDL(i-1))/2.0d0 dg2_TDL=(g2_TDL(i+1)-g2_TDL(i-1))/2.0d0 dg_TDL=dg1_TDL+dg2_TDL c write(9,*)'dg_TDL is ',dg_TDL if((dg_TDL).lt.1.d-18) then wtfrcn=wfc10 else wtfrcn=dg1_TDL/dg_TDL end if c write(9,*)'wtfractiof nonane is ',wtfrcn tempA=(wmuu0_TDL-t_TDL(i)/tstarcp/gammam/mssq_TDL)/rr0_TDL(i) tempB=tt0_TDL(i) c write(9,*)'T/T0_TDL is ',tt0_TDL(i) tempC=tt0_TDL(i)*wg_TDL tempF=fdhc(wtfrcn,t_TDL(i))/cp_TDL/t0-tt0_TDL(i)*wg_TDL c write(9,*)'latent heat of mixture is L = ',fdhc(wtfrcn,t_TDL(i)) tempD=(h_TDL- t_TDL(i)/tstarcp/gammam/mssq_TDL)/rr0_TDL(i)/tempF tempE=tt0_TDL(i)/tempF dlar_TDL=dg_TDL/tempE-tempD/tempE*dp c write(9,*)'dLN(A/A*) IS ',dlar_TDL dtt0_TDL=tempA*dp+tempB*dlar_TDL+tempC*dg_TDL dr_TDL=dp/tstarcp*t0/gammam/mssq_TDL-rr0_TDL(i)*dlar_TDL
ar_TDL(i+1)=ar_TDL(i-1)*dexp(2.0d0*dlar_TDL) rr0_TDL(i+1)=rr0_TDL(i-1)+2.0d0*dr_TDL tt0_TDL(i+1)=tt0_TDL(i-1)+2.0d0*dtt0_TDL wm_TDL=wmav0*dotncal/ & (dotncal-(dotncal-tN-tCH4)*fc_TDL(i)) ! Molecular weight for c c sound velocity ga_TDL=cp_TDL/(cp_TDL-8.3145/wm_TDL) ! Specific heat ratio for c sound velocity a_TDL=(ga_TDL*8.3145*t_TDL(i)/wm_TDL*1000.0)**0.5
if((w20+w30).gt.0.0)then fc_TDL(i+1)=(g1_TDL(i+1)+g2_TDL(i+1))/(w20+w30) ! Shinobu else fc_TDL(i+1)=0.0 end if c write(9,*)'fc_tdl is',fc_TDL(i+1),'in iteration number ',i+1 write(9,1105)xs(i),u(i)/100,t(i),g(i),t_is_s(i),t(i)- t_is_s(i), & pp0_is_s(i),pp0(i),fc(i), t_TDL(i),aratio(i),ar_TDL(i), & g1_TDL(i)/g1inf,g2_TDL(i)/g2inf,rr0_TDL(i),u_TDL(i)/100
264 write(10,1020)xs(i),u(i)/100,t(i),pp0(i),t_is_s(i),pp0_is_s(i), ! June05 Shinobu * (1-fc(i))*fcon/(allflux-fc(i)*fcon),g(i),fc(i),aratio(i), * rr0(i),tisd(i),pp0d(i) 1105 format(2f12.3,f8.2,e13.4,f8.2,f7.3,2e12.4,f8.4,f8.2,5f8.4,f8.2) 1020 format(f8.3,f10.2,f8.2,f8.4,f8.2,f8.4,2e13.4,f8.4,f8.4,f8.4, & f8.2,f8.4) !4pp plots 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)-t_is_s(i) and t(i)-tisd
do i = istart,ifin1-1 dtemp(i,kd) = t(i)-t_is_s(i) dt1 = t(i) - t_is_s(i) dt2 = t(i+1) -t_is_s(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 = pp0_is_s(i)+(0.5-dt1)/(dt2-dt1)*(pp0_is_s(i+1)- pp0(i)) ton = t(i)+(0.5-dt1)/(dt2-dt1)*(t(i+1)-t(i)) tiswon = t_is_s(i)+(0.5-dt1)/(dt2-dt1)*(t_is_s(i+1)- t_is_s(i)) else endif enddo write(*,*)'using the t-t_is_s = 0.5 k' write(*,1300)xon,pp0on*pct0,ton, & pp0ion*pct0,tiswon 1300 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,
& pp0on*pc10*pconv,pp0on*pc20*pconv,wetfil,xon 1301 format(f8.2,f8.2,f8.4,f8.2,f8.4,f8.4,f8.4,2x,a13,f7.1) 1302 format('@\"t0 p0 pct ton pon p1on p2on', & 6x,a13,'\"') xon=0.0 pp0on=0.0 ton=0.0 do i = istart,ifin1-1 dt1 = t(i) - tisd(i) dt2 = t(i+1) - tisd(i+1)
265 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)) pp0don = pp0d(i)+(0.5-dt1)/(dt2-dt1)*(pp0d(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)) else endif enddo write(*,*)'using the criterion t-tisd = 0.5 k' write(*,1300)xon,pp0on*(pc10+pc20)*pconv,ton, & pp0don*(pc10+pc20)*pconv,tisdon
write(*,*) 'finished integration' enddo enddo c now write out the dtemp files to dtemp.out
do i = istart,ifin1 write(13,1313)xs(i),(dtemp(i,j),j=1,ndata) enddo 1313 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 c regardless of the spacing c c m - order of the highest polynomial used in smoothing c n - number of y points in interval over which smoothing is performed c k - point whose smoothed value is desired c k0 - first point in set of n c sval - smoothed value returned to calling program c 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)
266 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(9,2)a goto 15 99 continue rewind 5 return 1 format(a100) 2 format(1x,a100) 3 format(1h1,20x,'input file',//) end *------chh22.02.01---* * c real function fdhc(dhc) c fdhc = dhc c return * real*8 function fdhc(wfc10,tk) double precision zc10,wfc10,tk,rg *------general nomenclature-*
267 c rg universal gas constant in units of c tk temperature of vapor condensing in kelvin c zc10 molar fraction of condensible 1 in vapor (zc10+zc20=1.0) *------condensible nomenclature-* c a2h2o - a4h2o h2o vapor pressure constants, wagner correlation c a1d2o - a6d2o d2o vapor pressure constants, c mwd2o d2o molecular weight c z d2o intermediate variable *------* double precision a1d2o,a2d2o,a3d2o,a4d2o,a5d2o,a6d2o,z,mwd2o !d2o pve constants double precision bbu,cbu,mwetod,dhcetod
rg=8.3145d0
*------* *-d2o clausius-clapeyron relation applied to equilibrium vapor pressure *-d2o valid for temperature range of 275-823K *-d2o hill, mcmillan, and lee, j. phys chem ref data, vol 11, no.1, p1-14 (1982) a1d2o= -7.81583d0 a2d2o= 17.6012d0 a3d2o=-18.1747d0 a4d2o= -3.92488d0 a5d2o= 4.19174d0 a6d2o=643.89d0 mwd2o=20.03d0
z=1-tk/a6d2o d2oa=a1d2o*z+a2d2o*z**1.9+a3d2o*z**2+a4d2o*z**5.5+a5d2o*z**10. d2ob=a1d2o+1.9d0*a2d2o*z**0.9+2.d0*a3d2o*z+5.5d0*a4d2o*z**4.5 &+10.d0*a5d2o*z**9. dhcd2o=-rg*(a6d2o*d2oa+tk*d2ob)/mwd2o
*------* *-BuOH clausius-clapeyron relation applied to equilibrium vapor pressure *-BuOH valid for temperature range of 243.2-303.2 K * And T. Schmeling and R. Strey, Ber. Bunsenges. Phys. Chem., vol 87, p871-874 (1983) c bbu= 9412.61d0 c cbu= 10.54d0
268 c mwbuOH=74.12d0 c dhcbuOH=rg*(bbu-cbu*tk)/mwbuOH c Use corrected equation 20 of Ruzicka and Majer J physical chem ref data 23, 1994 p 1-39 c Note the T is missing from the a1 term!Original units are J/mol a0 = 2.94690d0 a1 = -2.051933d-3 a2 = 1.903683d-6 Tb = 423.932d0 mwetod=128.260d0
Term1 = rg*exp(a0 + a1*tk + a2*tk*tk)
Term2 = Tb + tk*(tk - Tb)*(a1+2.0d0*a2*tk) ccc dhcetod= Hvap von Nonane!!!
dhcetod = (Term1*Term2)/mwetod
*------* fdhc = (wfc10*dhcetod)+(1.d0-wfc10)*dhcd2o c write(28,*)'dhc debug: fdhc= ',fdhc,' wfc10= ',wfc10,' tk= ',tk,' K' !debug dhc return
end
*------*
***************** Functions for Cp **********************************
real*8 function fcp2(tk,p,y1) double precision tk,p,y1 double precision mw,a0,a1,a2 c** Cp of BuOH ****** p: Total static pressure ****** y1: Mole fraction of condensable 1 in vapor phase c mw=74.12 c a0=30.941d0 c a1=0.10037d0 c a2=7.322d-5 c fcp2=(a0+a1*tk+a2*tk*tk)/mw (data for EtOD) c fcp2=1.473 used a const cp for BuOh?? c [J/g*K]
269 *----Cp Nonane--Ruzicka & Majer J.Phys.Chem.Ref.Data.Vol.23.No.1,1994 ------***NEU machen * taken from Bures et al, Chem Eng Sci, 36, 529-537 (1981) c------Einheit J/molK------
A = 148.15036d0 B1 = 288.24904d0 B2 = 178.57491d0 C1 = 1380.8003d0 C2 = 3051.1566d0
Term1 = (C1/tk)*(C1/tk)*dexp(-C1/tk)/(1.0d0 - dexp(- C1/tk))**2
Term2 = (C2/tk)*(C2/tk)*dexp(-C2/tk)/(1.0d0 - dexp(- C2/tk))**2
cpnonane = A + B1*Term1 + B2*Term2
mw=128.260d0
fcp2 = cpnonane/mw c unit for the code should be J/g*K
return end
real*8 function fcp3(tk,p,y2) double precision tk,p,y2 double precision rg,mw,a0,a1,a2,a3 c** Cp of D2O: Fitted to the data in JCP 22, 2051 (1954) at T= 160- 340 K rg=8.3145 mw=20.03 a0=4.1712 a1=-2.2388d-3 a2= 8.6096d-6 a3=-5.6304d-9 fcp3=rg*(a0+a1*tk+a2*tk**2+a3*tk**3)/mw c fcp3=1.710d0 return end
real*8 function fcp4(tk,p,y3) double precision tk,p,y3 double precision rg,mw,a0,a1,a2,a3
270 c** Cp of CH4: Fitted to the data in J. Chem. Eng. Data 8, 547 (1963) at T= 160-340 K rg=8.3145 mw=16.04 a0=4.337 a1=-3.7677d-3 a2= 9.1107d-6 a3= 1.0178d-9 fcp4=rg*(a0+a1*tk+a2*tk**2+a3*tk**3)/mw c fcp4=2.226d0 return end
************** Temperature derivative of Cp ****************
real*8 function fdcp2dt(tk,p,y1) double precision tk,p,y1,mw,a1,a2,c1,c2 c mw=74.12 c a1=0.10037d0 c a2=7.322d-5 c fdcp2dt=(a1+2.d0*a2*tk)/mw c----Cp Nonane--Ruzicka & Majer J.Phys.Chem.Ref.Data.Vol.23.No.1,1994 ------***NEU machen * taken from Bures et al, Chem Eng Sci, 36, 529-537 (1981) c------Einheit J/molK------
A = 148.15036d0 B1 = 288.24904d0 B2 = 178.57491d0 C1 = 1380.8003d0 C2 = 3051.1566d0 mwnonane=128.258d0 ccc Näherung für cp/dt
tk1 = tk+ 0.01d0 tk2 = tk - 0.01d0
Term1 = (C1/tk1)*(C1/tk1)*dexp(-C1/tk1)/ & (1.0d0 - dexp(-c1/tk1))**2 Term2 = (C2/tk1)*(C2/tk1)*dexp(-C2/tk1)/ & (1.0d0 - dexp(-c2/tk1))**2
cpnonane1 = (A + B1*Term1 + B2*Term2)/mwnonane
Term1 = (C1/tk2)*(C1/tk2)*dexp(-C1/tk2)/ & (1.0d0 - dexp(-c1/tk2))**2 Term2 = (C2/tk2)*(C2/tk2)*dexp(-C2/tk2)/ & (1.0d0 - dexp(-c2/tk2))**2
cpnonane2 = (A + B1*Term1 + B2*Term2)/mwnonane
271 fdcp2dt = (cpnonane1-cpnonane2)/(0.02d0)
return end
real*8 function fdcp3dt(tk,p,y2) double precision tk,p,y2 double precision rg,mw,a1,a2,a3 c** Cp of D2O: Fitted to the data in JCP 22, 2051 (1954) at T= 160- 340 K rg=8.3145 mw=20.03 a1=-2.2388d-3 a2= 8.6096d-6 a3=-5.6304d-9 fdcp3dt=rg*(a1+2.d0*a2*tk+3.d0*a3*tk**2)/mw c fdcp3dt=0.0 return end
real*8 function fdcp4dt(tk,p,y3) double precision tk,p,y3 double precision rg,mw,a1,a2,a3 c** Cp of CH4: Fitted to the data in J. Chem. Eng. Data 8, 547 (1963) at T= 160-340 K rg=8.3145 mw=16.04 a1=-3.7677d-3 a2= 9.1107d-6 a3= 1.0178d-9 fdcp4dt=rg*(a1+2.d0*a2*tk+3.d0*a3*tk**2)/mw c fdcp4dt=0.0 return end
*************** Pressure derivative of Cp *********************
real*8 function fdcp2dp(tk,p,y1) double precision tk,p,y1 fdcp2dp=0.d0 return end
real*8 function fdcp3dp(tk,p,y2) double precision tk,p,y2 fdcp3dp=0.d0 return end
272 real*8 function fdcp4dp(tk,p,y3) double precision tk,p,y3 fdcp4dp=0.d0 return end
***************** Functions for Cpl ********************************** c** Cpl of Nonane real*8 function fcpl2(tk) double precision tk fcpl2=1.1061d-5*(tk**2)-2.8403d-3*(tk)+2.07130d0 return end c** Cpl of D2O real*8 function fcpl3(tk) double precision tk fcpl3=4.205d0 return end
******** Temperature derivative of gamma of gas mixture ***
real*8 function dgdt(tk,p,y1,y2,y3,w20,w30,w40,wmav0) double precision tk,p,y1,y2,y3,w20,w30,w40,wmav0 double precision rg,cpN,cp,gamma,dcpdt
rg=8.3145 cpN=1.0397
cp= (1.d0-w20-w30-w40)*cpN+w40*fcp4(tk,p,y3) & +w20*fcp2(tk,p,y1)+w30*fcp3(tk,p,y2) gamma=cp/(cp-rg/wmav0)
dcpdt= w20*fdcp2dt(tk,p,y1)+w30*fdcp3dt(tk,p,y2) & +w40*fdcp4dt(tk,p,y3)
dgdt=gamma*(1.d0-gamma)/cp*dcpdt
return end
******** Pressure derivative of gamma of gas mixture ******
273 real*8 function dgdp(tk,p,y1,y2,y3,w20,w30,w40,wmav0) double precision tk,p,y1,y2,y3,w20,w30,w40,wmav0 double precision rg,cpN,cp,gamma,dcpdp
rg=8.3145 cpN=1.0397
cp= (1.d0-w20-w30-w40)*cpN+w40*fcp4(tk,p,y3) & +w20*fcp2(tk,p,y1)+w30*fcp3(tk,p,y2) gamma=cp/(cp-rg/wmav0)
dcpdp= w20*fdcp2dp(tk,p,y1)+w30*fdcp3dp(tk,p,y2) & +w40*fdcp4dp(tk,p,y3)
dgdp=gamma*(1.d0-gamma)/cp*dcpdp
return end
274 Appendix B Thermo-physical Properties of D2O, Nonane and Nitrogen
275 Table B 1 Thermophysical properties of D2O.
Thermophysical Ref property
Molar weight 0.02003
-1 MD2O (kg mol )
Critical temperature 643.89 1
Tc (K)
Critical Pressure, 21.66 1
pc (MPa)
Vapor Properties
Specific isobaric heat 3 2,3 C pD2O(v) R(4.171 2.239 10 T £ capacity 6 2 9 3 -1 -1 8.610 10 T 5.360 10 T ) C pD2O(v) (J mol K ) for 160 T / K 340
Liquid Properties
Equilibrium vapor pressure p T 1,4 ln e c ( 1.9 2 5.5 10 ) p T 1 2 3 4 5 pe c
1 7.81583 2 17.6012
3 18.1747 4 3.92488
4.19174 5 continued…
276 Table B1 continued…
Latent heat of L RT (ln( pe / pc ) ) / M D2O condensation* 1.9 0.9 2 5.5 4.5 10 9 L (J g-1) 1 2 3 4 5
Density, 0.33 5 T 231 Tc T l 0.09 tanh 0.847 0.338 -3 51.5 T ρl (g cm ) c
Compressibility 1011 (a bt ct 2 dt 3 et 4 ft 5 ) 6
1 (Pa ) a = 53.5216 d = 8.5541 10-5
b =0.4536 e = 5.4089 10-7
c =8.712 10-3 f = 1.3478 10-9
Scattering length density 6 SLD 9.110 with respect to X-rays
-2 SLD (Å )
Vapor-liquid surface 2 7 vl 93.6635 0.009334T 0.000287T -1 tension vl (mN m )
*The Clausius-Clapeyron equation was applied to the equilibrium vapor pressure curve from Hill et al.1, 4
£ The isobaric heat capacity of D2O vapor is assumed to be constant between 210-240 K to be 33.55 Joule/mol/K.
277 Table B 2 Thermo-physical properties of nonane.
Thermophysical Ref property
Molar weight 0.12823 8 -1 Mnonane (kgmol )
Critical 594.55 8 temperature
Tc (K) Critical Pressure, 2.29 8
pc (MPa)
Vapor Properties
Specific isobaric C pnonane 148.15036 9 heat capacity 2 (1380.8003 / T ) exp(1380.8003 / T ) -1 -1 288.24904 2 Cp-nonane (Jmol K ) [1 exp(1380.8003 / T )] (3051.1566 / T ) 2 exp(3051.1566 / T ) 178.57491 [1 exp(3051.1566 / T )]2
Liquid Properties
Equilibrium vapor pe 101.325exp[(1 423.932 / T ) exp(2.94690 9 pressure 2.05193310 3 T 1.903683910 6 T 2 )]
pe (kPa)
Latent heat of 3 9 L R / M nonane exp(2.94690 2.05193310 T condensation€ 1.90368310 6 T 2 )[423.932 T (T 423.932) 3 6 L (J g-1) (2.05193310 3.80736610 T ) continued…
278 Table B2 continued…
Density, 1/ 3 2 / 3 10 l 0.238[11.927780(1T / Tc ) 0.9302189(1T / Tc ) 4 / 3 -3 1.334128(1T / T ) 1.392823(1T / T ) ρl (g cm ) c c
Scattering length 6 SLD 7.5 10 density with respect to X-rays
-2 SLD (Å )
Vapor-liquid 2 11 vl 24.84 9.417 10 (T 273.15) surface tension
-1 vl (mN m )
The compressibility for liquid nonane, K over 200< T/ K < 240 was estimated as 710-10 Pa-1 (NIST 2002).
279 Table B 3 Thermo-physical properties of N2.
Nitrogen Ref.
-1 Molar weight, Mw (kg mol ) 0.0280135 12
Specific isobaric heat capacity, 1.0395 13-15
-1 -1 Cp (J mol K )
Scattering length density with respect to X-rays 9 SLD 8.9 10
-2 SLD (Å )
References
1. Hill, P. G.; MacMillan, R. D. C.; Lee, V. A Fundamental Equation of State for Heavy Water. Journal of Physical and Chemical Reference Data 11, 1-14 (1982).
2. Tanimura, S.; Wyslouzil, B. E.; Wilemski, G. CH3CH2OD/D2O Binary Condensation in a Supersonic Laval Nozzle: Presence of Small Clusters Inferred from a Macroscopic Energy Balance. Journal of Chemical Physics 132, 1-22 (2010).
3. Abraham, S. F.; Lester, H. High-Speed Machine Computation of Ideal Gas Thermodynamic Functions. I. Isotopic Water Molecules. Journal of Chemical Physics 22, 2051-2058 (1954).
4. Hill, P. G.; Chris, R. D. Saturation States of Heavy Water. Journal of Physical and Chemical Reference Data 9, 735-750 (1980).
280 5. Wölk, J.; Strey, R. Homogeneous Nucleation of H2O and D2O in Comparison: The Isotope Effect. Journal of Physical Chemistry B 105, 11683-11701 (2001).
6. Frank, J. M.; Fred, K. L. Isothermal Compressibility of Deuterium Oxide at Various Temperatures. Journal of Chemical Physics 54, 946-949 (1971).
7. Joseph, J. J. The Surface Tension of Pure Liquid Compounds. Journal of Physical and Chemical Reference Data 1, 841-1010 (1972).
8. D.R. Lide. Handbook of Chemistry and Physics, 84th ed(CRC, Boston, 2003).
9. K. Růžička and V. Majer. Simultaneous Treatment of Vapor Pressures and Related Thermal Data between the Triple and Normal Boiling Temperatures of n-
Alkanes C5-C20. Journal of Physical and Chemical Reference Data 2, 1-39 (1994).
10. Cibulka. Saturated liquid densities of 1-alkanols from C1 to C10 and n-alkanes
from C5 to C16. A critical evaluation of experimental data. Fluid Phase Equlibria 89, 1-18 (1993).
11. M.M. Rudek; J. Fisk; V.M. Chakarov and J. Katz. Condensation of a supersaturated vapor. XII. The homogeneous nucleation of the n-alkanes. Journal of Chemical Physics 105, 4707-4713 (1996).
12. Coplen, T. B. Atomic Weights of the Elements 1995 (Reprinted from International Union of Pure and Applied Chemistry from Pure Appl. Chem. vol 68, pg 2339, 1996). Journal of Physical and Chemical Reference Data 26, 1239- 1253 (1997)
13. Span, R.; Lemmon, E. W.; Jacobsen, R. T.; Wagner, W.; Yokozeki, A. A Reference Equation of State for the Thermodynamic Properties of Nitrogen for Temperatures from 63.151 To 1000 K and Pressures to 2200 MPa. Journal of Physical and Chemical Reference Data 29, 1361-1433 (2000).
281 14. Benedict, M. Pressure, Volume, Temperature Properties of Nitrogen at High Density. II. Results Obtained by a Piston Displacement Method. Journal of American Chemical Society 59, 2233-2242 (1937).
15. Mage, D. T.; Jones, J. M. L.; Katz, D. L.; Roebuck, J. R. Experimental Enthalpies For Nitrogen. Chemical Engineering Progress Symposium Series No. 44, 59, 61 (1963).
16. NIST (2002). NIST Standard Reference Database REFPROP 23, in Reference Fluid Thermodynamic and Transport Properties.
282 Appendix C:Appendix to Chapter 3:Droplet Temperatures and
Growth Rate Calculations for Pure Nonane and Pure D2O Nanodroplets
283 Figure C 1 (a) and (b) The experimental and theoretical droplet temperatures for nonane at pv0 =489 Pa are calculated as described in the manuscript. The error bars on Td,exp are based on a 5% uncertainty in calculating the number density and a 5% uncertainty in calculating the condensation rate, dg/dt based on the subjectivity of the sigmoidal fit.
Td,exp is consistently higher than Td,exact and the absolute difference between them decreases as we move downstream. (c) The experimental and theoretical growth rates for nonane at pv0 =489 Pa. Even when Td is much higher than T the isothermal and non- isothermal growth laws predict essentially the same growth rate.
284 Figure C 2 (a) and (b) The experimental and theoretical droplet temperatures for nonane at pv0 =322 Pa are calculated as described in the manuscript. The error bars on Td,exp are based on a 5% uncertainty in calculating the number density and a 5% uncertainty in calculating the condensation rate, dg/dt based on the subjectivity of the sigmoidal fit.
∆Texp is almost twice that of ∆Texact and this pv0 represents our worst disagreement. The
∆Texp at 2 µs is ~50 K and we think that it is a faulty reading and have not shown it here.
(c) The experimental and theoretical growth rates for nonane at pv0 =322 Pa. Even when
Td is much higher than T the isothermal and non-isothermal growth laws predict essentially the same growth rate.
285 260 (a) 250 240 (K) d
T 230 220 210 (b)
Texp 30 Texact with qe=1
Texact with qe=0.5
20 (K) T - d T
10
0 0 20 40 60 80 100 time since onset (s)
Figure C 3 (a) and (b) The experimental and theoretical droplet temperatures for D2O at pv0 =520 Pa are calculated as described in Section 3.3.2. The error bars on Td,exp are based on a 5% uncertainty in calculating the number density and a 5% uncertainty in calculating the condensation rate, dg/dt based on the subjectivity of the sigmoidal fit.
286 Appendix D: Appendix to Chapter 4: The Experimental Parameters of Binary Traces
287 Figure D 1 The experimental parameters for a case when nonane and D2O are condensing together (pv0= 357 Pa D2O
+319 Pa nonane). a) The pressure and temperature along with the isentropic profiles. b) The average radius r and the spread σ in the droplet size distributions. c) The mass fraction condensed (g). The black symbols are g calculated from the FTIR vapor analysis. The white symbols are g calculated from the FTIR analysis from the liquid part of the spectrum. The dashed line is the incoming mass fraction of the vapor, gmax. The dark lines are gfit as mentioned in the integrated analysis section 2.3.4.d) D2O dominates nucleation in this case and the D2O nucleation pulse calculated from CNT is depicted by grey line and is normalized by its maximum value and is depicted on the left axis. The number densities calculated from SAXS are depicted along with the scaled nucleation values calculated from integrating the ~ CNT nucleation rates to match the maximum value of N SAXS . The data presented in this figure was first published in AIP conference Proceedings, Volume 157, pages-51-54 (2013).
288 Figure D 2 The experimental parameters when both nonane (pv0= 487 Pa) and D2O (pv0= 364 Pa) are condensing together. The symbols are explained in Figure D1. Since this is a case of competitive nucleation, the nucleation pulse is not shown here.
289 Figure D 3 The experimental parameters when both nonane (pv0= 311 Pa) and D2O (pv0= 538 Pa) are condensing together. The symbols are explained in Figure D1. This is a case where D2O dominates nucleation.
290 Figure D 4 The experimental parameters when both nonane (pv0= 622 Pa) and D2O (pv0= 547 Pa) are condensing together. The symbols are explained in Figure D1. This is a case where D2O dominates nucleation.
291 Figure D 5 The experimental parameters when both nonane (pv0= 325 Pa) and D2O (pv0= 704 Pa) are condensing together. The symbols are explained in Figure D1. This is a case where D2O dominates nucleation.
292 Figure D 6 The experimental parameters when both nonane (pv0= 489 Pa) and D2O (pv0= 714 Pa) are condensing together. The symbols are explained in Figure D1. This is a case where D2O dominates nucleation.
293 Figure D 7 The experimental parameters when both nonane (pv0= 621 Pa) and D2O (pv0= 718 Pa) are condensing together. The symbols are explained in Figure D1. This is a case where D2O dominates nucleation.
294 Table D 1 The exact pv0 of nonane and D2O for different experiments where nonane and D2O vapors condense together ~ with their mass fractions gmax which enter the nozzle. NSAXS is the maximum number density of the aerosol measured by SAXS before coagulation, ΔtJmax is the characteristic nucleation time and Jmax are the measured nucleation rates.
SD2O and Snonane are the saturation ratios reached by nonane and D2O respectively at Jmax.
~ pv0 pv0 gmax gmax 18 ΔtJmax Jmax SD2O at Snonane NSAXS 10 nonane D2O nonane D2O Jmax at Jmax -1 (Pa) (Pa) (kg ) (μs) (#/cm-3s-1)
322 0 0.0470 0 4.2 21.3 2.0 1016 - 4670
489 0 0.0700 0 2.0 18.7 1.2 1016 - 2010
625 0 0.0881 0 1.4 17.7 9.4 1015 - 1350
0 346 0 0.00822 30 14.3 2.5 1017 188 -
0 520 0 0.0124 32 10.7 3.8 1017 124 -
0 683 0 0.0162 32 10.6 4.1 1017 96.6 -
319 357 0.0468 0.00817 15 14.3 1.2 1017 187 1300
311 538 0.0457 0.0124 16 12.9 1.5 1017 125 520
325 704 0.0477 0.0162 15 10.8 1.8 1017 105 335
487 364 0.0699 0.00817 7.4 14.4 5.8 1016 163 1670
487 538 0.0701 0.0121 9.9 12.5 9.6 1016 121 786
489 714 0.0704 0.0161 10 11.4 1.2 1017 99.0 466
618 367 0.0875 0.00813 2.2 16.9* 1.5 1016 95.0 1160
622 547 0.0881 0.0121 6.5 13.2 5.9 1016 111 900
621 718 0.0882 0.0159 7.9 10.6 9.4 1016 103 611
*The characteristic time for nucleation ΔtJmax in all binary cases is calculated by
295 J dt D2O tJ max except in the case denoted by * where it is calculated by J max,D2O
J dt nonane tJ max J max,nonane
296 Appendix E: Appendix to Chapter 5: Analytical expressions of the Form factor and Guinier analysis for the Lens-on-Sphere Structure
297 Figure E1 is an illustration for a lens-on-sphere structure. The sphere has a radius R2 and the spherical lens has aradius R1. The distance between the centers of the lens and sphere is d. θc is the angle of contact of the lens on the sphere. The scattering length density of the sphere is ρb2, for the lens it is ρb1 and that of the surrounding is ρb3. The scattering length density differences are defined as
b1 b1 b3 (E1)
b2 b2 b3 (E2)
The distance between the centers of the lens and the sphere is defined by
2 2 2 d R1 R2 2R1R2 cos c (E3)
c R R 1 2 d
Figure E 1 An illustration of the lens-on-sphere structure
E.1 Form factor for the lens-on-sphere structure
The form factor for the lens on sphere structure (Wilemski et al., 2013) is given by
P (q) f 2 f [C ( ) C ( )]sind j j2 b1 j2 1 2 0 (E4) 2( ) 2 [(C ( ) C ( )) 2 (S ( ) S ( )) 2 ]sind b1 1 2 1 2 0
298 The terms on the right hand side of Equation (E4) represent scattering from the various parts of the lens-on-sphere structure. The first term, fj2 is the scattering due to the D2O sphere. The third term represents scattering from the nonane lens and the second term represents the interaction term between the D2O sphere and the nonane lens. The expression for fj2 is given by
sin(qR2 ) qR2 cos(qR2 ) f j2 (q) 3 b2V2 3 (E5) (qR2 ) where V2 is the volume of the D2O sphere and q is the scattering vector. The expressions in Equation (E4) are given by
R1 1 2 2 C1 ( ) R1 z J1 (u1 )cos[q(z d)]dz (E6) 2 q 1 z1
R2 1 2 2 C2 ( ) R2 z J1 (u2 )cos[qz]dz (E7) 2 q 1 z2
R1 1 2 2 S1 ( ) R1 z J1 (u1 )sin[q(z d)]dz (E8) 2 q 1 z1
R2 1 2 2 S 2 ( ) R2 z J1 (u2 )sin[qz]dz (E9) 2 q 1 z2
Here, J1 is the Bessel function of the first order,
cos (E10)
2 2 2 u2 q 1 R2 z (E11)
2 2 2 u1 q 1 R1 z (E12)
The lower limits of integration in equations (E6) to (E9) are given by
299 z1 R1 cos (E13)
z2 d R1 cos R2 cos (E14) where
cos (R2 cos c R1 ) / d (E15)
cos (R2 R1 cos c ) / d (E16)
E.2 Guinier analysis for the lens-on sphere structure
In Guinier analysis, we look at the behavior of the scattering Intensity I(q) at small values 2 2 of scattering vector q. The plot of Ln(I) vs q is linear with a slope proportional to rG and intercept proportional to N. The Guinier plot equation in the region where qrG<<1 is
2 2 ln I(q) ln I(0) q rG / 3 (E17)
We derive the explicit expressions for I(0) for a droplet of size l which is given by
(E18) where
(E19)
(E20)
(E21)
300 (E22)
After substituting R1=γR2 and skipping all intermediate steps, we obtain
(E23) where
(E24)
The angle brackets define an average using the Schulz probability density function.
Equation (E23) looks like an expression for scattering from polydisperse spheres with an effective scattering length bI that weakly depends on the droplet radii.
We also derive the radius of gyration rG for this model and it is given by
(E25) where the function M(α) is defined as
M (4 5cos cos5 ) /8 (E26)
After simplifying for I(0), we get the following expression for rG
301 8 2 3 R2 r 2 bG (E27) G 5 6 2 R2 bI where
(E28)
References:
1. Wilemski, G., Obeidat, A. and Hrahsheh, F. Form factors for Russian doll droplet models. AIP Conference Proceedings 1527, 144-147 (2013).
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