Nucleation of N-Nonane in Mixtures of Methane, Propane, and Carbon Dioxide

Nucleation of N-nonane in mixtures of methane, propane, and carbon dioxide

Citation for published version (APA): Labetski, D. G. (2007). Nucleation of N-nonane in mixtures of methane, propane, and carbon dioxide. Technische Universiteit Eindhoven. https://doi.org/10.6100/IR624554

DOI: 10.6100/IR624554

Document status and date: Published: 01/01/2007

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Download date: 02. Oct. 2021 NUCLEATION OF N-NONANE IN MIXTURES OF METHANE, PROPANE, AND CARBON DIOXIDE Copyright c 2007 D. Labetski

Omslagontwerp: P. Verspaget, D. Labetski Druk: Universiteitsdrukkerij, TUE

CIP-DATA LIBRARY TECHNISCHE UNIVERSITEIT EINDHOVEN

Labetski, Dzmitry

Nucleation of n-nonane in mixtures of methane, propane, and carbon dioxide / by Dzmitry Labetski. - Eindhoven : Technische Universiteit Eindhoven, 2007. - Proefschrift. - ISBN 978-90-386-2222-4 NUR 910 Trefw.: condensatie / druppelvorming / gasdynamica / aardgas. Subject headings: condensation / nucleation / gas dynamics / natural gas. NUCLEATION OF N-NONANE IN MIXTURES OF METHANE, PROPANE, AND CARBON DIOXIDE

PROEFSCHRIFT

ter verkrijging van de graad van doctor aan de Technische Universiteit Eindhoven, op gezag van de Rector Magniﬁcus, prof.dr.ir. C.J. van Duijn, voor een commissie aangewezen door het College voor Promoties in het openbaar te verdedigen op donderdag 22 maart 2007 om 16.00 uur

door

Dzmitry Labetski

geboren te Smolevichi, Belarus Dit proefschrift is goedgekeurd door de promotoren:

prof.dr.ir. M.E.H. van Dongen en prof.dr.ir. A. Hirschberg

This research was ﬁnancially supported by Twister B.V. and by the Dutch Technol- ogy Foundation STW, grant ESF.6472. Facts do not cease to exist because they are ignored.

Aldous Huxley

Contents

1 Introduction 1 1.1 Thermodynamics and kinetics of homogeneous nucleation ...... 2 1.2 Experimental studies on homogeneous nucleation ...... 5 1.3 Application ...... 7 1.4 Thesisoverview...... 8 References ...... 9

2 Wave tube experimental method 11 2.1 Nucleation pulse method ...... 12 2.2 Pulse expansion wave tube ...... 13 2.3 Determination of pulse conditions: pressure, temperature, time duration...... 17 2.4 Detection of macroscopic droplets ...... 22 2.5 Mixture preparation and initial mixture compositions ...... 31 2.6 Experimental procedure ...... 36 References ...... 37

3 Nucleation and droplet growth 39 3.1 Kinetics of homogeneous nucleation ...... 39 3.2 Nucleationtheories ...... 48 3.3 Nucleationtheorem...... 54 3.4 Dropletgrowth ...... 58 References ...... 68

4 Gradient-theory computation of surface tension and nucleation rate for n-nonane clusters 71 4.1 Theory ...... 72 4.2 Surface tension and the work of formation ...... 77 4.3 Nucleationrate ...... 81 4.4 Comparisons...... 83 4.5 Conclusions ...... 86 4.6 Thermophysical properties of n-nonane ...... 86 References ...... 87

5 Results and discussion 91 5.1 n-Nonane nucleation in methane ...... 91 5.2 n-Nonane nucleation and droplet growth in methane/propane car- riergasmixtures...... 99 5.3 n-Nonane nucleation and droplet growth in methane/carbon diox- idemixtures ...... 104 References ...... 109

vii CONTENTS

6 Conclusions and recommendations 111

A Phase equilibrium and surface tension in multicomponent mixtures 115 A.1 Phase equilibrium in two-phase system ...... 115 A.2 Equationsofstate ...... 116 A.3 Supersaturation of vapor diluted in the carrier gas ...... 117 A.4 Surface tension in mixtures ...... 117 References ...... 118

B Simultaneous nucleation and droplet growth in the expansion wave tube119 B.1 Characteristic depletion time ...... 119 B.2 Simultaneous nucleation and droplet growth model ...... 121 B.3 Comparison of model predictions with experimental data ...... 122 References ...... 125

C Experimental data 129

Summary 137

Samenvatting 141

Acknowledgments 145

Curriculum Vitae 147

viii Chapter 1

Introduction

Condensation is a phenomenon that is a common-life experience. Imagine your- self taking a milk bottle (or a beer bottle if you prefer) from your refrigerator. Left untouched, after some time the bottle will be covered with tiny droplets. Wait a little more and droplets will grow to bigger sizes. What happens is that the water vapor present in the air contacts with the bottle surface and is cooled down. At this lower temperature it cannot exist in the gas phase and has to transform into liquid. So, condensation is a transition process from one phase to another. Because condensation takes place on a surface it is called heterogeneous condensation. Heterogeneous condensation is a phase transition process which occurs on foreign particles such as ions, dust particles, or at some solid surface.

Fig. 1.1: Water vapor condensation near the open refrigerator. CHAPTER 1. INTRODUCTION

If you are curious, you may continue "experimenting". Put the bottle back into the refrigerator and and turn the knob to decrease the temperature inside. After a while, take the bottle out, put it on the table and observe what is going to happen. You will see that now the condensation is more intense, more droplets are formed on the surface and they are growing faster. From this observation, you can conclude that the more remote the vapor state from its liquid state at the bottle surface the more intense the condensation is. In this particular case, the remoteness from equilibrium or the vapor supersaturation can be characterized by the temperature difference between two states. Let us continue to decrease the temperature, at some point the temperature will be so low that when you open the refrigerator the condensation will start at once. The cold air from the refrigerator will mix with air in the room, cool it down and you will see the formation of very small droplets which manifest themselves as a dense white mist (Fig. 1.1). What you just have observed is a combination of heterogeneous condensation on aerosols and homogeneous condensation – the vapor supersaturation may become so high that foreign particles or surfaces are not always needed. The spontaneous gathering of a few vapor molecules will play the role of a droplet growth nucleus. Because the condensation is not limited by the presence of foreign particles, homogeneous condensation may become a very intense process. The droplet formation rate or nucleation rate strongly depends on supersaturation. In this thesis only homogeneous nucleation will be considered and studied.

1.1 Thermodynamics and kinetics of homogeneous nucleation

When the vapor is brought into a thermodynamically unstable state, it adjusts itself to a new condition through a transition process – condensation. During condensation, the nuclei or clusters of the new condensed phase are formed and then these clusters start to grow forming macroscopic droplets, and eventually the bulk liquid phase.

The energy of cluster formation is a key parameter in the nucleation process. In the simple case of unary nucleation, the energy of cluster formation consists of bulk and liquid terms [1, 2]:

W = n(µgas µeq) + aσn2/3, (1.1) − − where n is the number of molecules in the cluster, µgas is the actual chemical potential of the gas phase and µeq that of the liquid (and vapor) phase at equilibrium, a is the cluster surface per molecule, and σ is the surface tension. The bulk term, the ﬁrst in the right hand side (rhs), characterizes how much energy is gained when

2 1.1. THERMODYNAMICS AND KINETICS OF HOMOGENEOUS NUCLEATION vapor molecules move from the gas phase to the bulk liquid phase. The difference in chemical potentials indicates how supersaturated the vapor is. To have a quantitative value for the supersaturation it can be deﬁned as

µgas µeq S = exp − . (1.2) k T b The surface term, the second in the rhs, tells how much energy is needed to build the cluster surface.

For a supersaturated vapor, the bulk and surface terms in (1.1) have different signs and different dependencies on the cluster size n. As a result, the energy of cluster formation has a distinctive maximum as a function of cluster size: when the cluster is small, the surface term prevails, which is proportional to n2/3, but as the cluster is growing the bulk term proportional to n sooner or later will dominate the surface term. As an example, the dependence of cluster formation energy as a function of cluster size is shown in Fig. 1.2 for different values of the supersaturation for n-nonane vapor. In equilibrium, the supersaturation equals unity and the energy of cluster formation is a steadily increasing function of cluster size. For a supersaturated vapor, the maximum in the energy of cluster formation appears at some value of the cluster size called the critical size. Then, the cluster is called a critical cluster. The clusters with sizes less than critical have a tendency to decrease its energy by evaporation. The clusters with sizes larger than critical decrease their energy by catching more vapor molecules and growing to macroscopic sizes. In the example in Fig. 1.2, at S = 50 the critical size n∗ is 25 molecules, and n∗ = 15 for S = 100. The critical size as well as the maximum energy of formation are decreasing functions of supersaturation. Therefore, the higher the supersaturation the less the energy and number of molecules needed to form a critical cluster.

From (1.1) the size n∗ and the formation energy W ∗ of the critical cluster can be found for given S: 8 aσ 3 n∗ = (1.3) 27 ∆µ and 1 W ∗ = n∗ ∆µ, (1.4) 2 with ∆µ µgas µeq. Apparently, a simple relation exists between the three ≡ − important variables W ∗, n∗ and ∆µ which follows from (1.3) and (1.4):

dW ∗ = n∗. (1.5) d∆µ −

Kashchiev [3] has shown that (1.5) is of general validity and can be applied to a very wide range of phase transitions, not only to unary vapor-liquid nucleation. It also plays an important role in the analysis of nucleation experimental data, because it allows to derive the size of the critical cluster from macroscopic mea-

3 CHAPTER 1. INTRODUCTION

100 S = 1 S = 50 S = 100 80

60 T)

b 40 W/(k

-20 1 10 100 n

Fig. 1.2: Formation energy W of n-nonane clusters as a function of cluster size n at 240 K and 10 bar. The higher is the supersaturation S the lower is the energy barrier and the corresponding critical size.

surements.

The expression (1.2), deﬁning supersaturation, can be simpliﬁed for two often encountered practical cases: ideal pure vapor and a dilute vapor in a real carrier sat gas. For an ideal vapor, ∆µ is kbT ln(P/P ), and therefore

P S = , (1.6) P sat where P is the pressure of the supersaturated vapor and P sat is the vapor pressure at phase equilibrium. For a dilute vapor in a carrier gas, (1.2) can be simpliﬁed (see Sec. A.3) to y S = , (1.7) yeq where y is the vapor molar fraction in the supersaturated state and yeq is the vapor molar fraction at phase equilibrium at the given total pressure. Supersaturation has a simple physical meaning, it measures to what extent the number of vapor molecules in the supersaturated state exceeds the number of vapor molecules at equilibrium.

The intensity of condensation is characterized by the nucleation rate – the number of droplets formed per unit volume per unit time. A very rude nucleation rate expression can be obtained if we assume that a droplet is formed when a critical cluster catches a single vapor molecule. Such a supercritical cluster is on the right side of the energy maximum (see Fig. 1.2), so it will decrease its energy by growing and eventually becoming a macroscopic droplet. The number density of critical

4 1.2. EXPERIMENTAL STUDIES ON HOMOGENEOUS NUCLEATION

clusters ρ∗ can be written in the form of a Boltzmann-type relation:

W ∗ ρ∗ = ρ exp , (1.8) 1 −k T b where ρ1 is the number density of monomers in the gas phase. The impingement rate ζ of monomers with the critical cluster can be written on the basis of kinetic theory as 1/2 2/3 kbT ζ = an∗ ρ . (1.9) 1 2πM Combining these two expressions yields a simple estimate for the nucleation rate J: 1/2 2/3 2 kbT W ∗ J = an∗ ρ exp . (1.10) 1 2πM −k T b The size and the formation energy of the critical cluster are functions of the chemical potential difference ∆µ or supersaturation S. An example of the nucleation rate dependence on supersaturation for n-nonane is shown in Fig. 1.3. Obviously, the nucleation rate strongly depends on supersaturation: an increase of supersaturation by a factor 2 gives about 5 orders of magnitude increase in the nucleation rate.

1020

1018

) 10 -1 s -3

J (m 1014

1012

1010 20 40 80 160 320

Sc9

Fig. 1.3: n-Nonane nucleation rate J as a function of supersaturation Sc9 at 240 K and 40 bar.

1.2 Experimental studies on homogeneous nucleation

There is a variety of experimental methods and techniques to study homogeneous nucleation in gas-vapor mixtures. A full survey of experimental methods and a critical analysis has been given among others by Springer [4], Heist and He [5].

5 CHAPTER 1. INTRODUCTION

The most important characteristics of a particular experimental setup are the range of achievable nucleation rates together with the operational range of temperatures, pressures, and vapor supersaturation. By combining different methods, it is possible to cover a large range of nucleation rates. For example, the four widely used techniques: the diffusion cloud chamber, the piston cloud chamber, the expansion- wave tube, and the supersonic nozzle, all together cover a range of nucleation rates 4 20 1 3 from 10 to 10 s− m− .

During nucleation, clusters of the condensed phase are formed. For typical experimental conditions, these clusters contain ten to hundred vapor molecules. But what is actually detected in the experiments are macroscopic droplets with sizes of the order of billions of molecules. This poses a problem to any experimental technique: to ensure that the nucleation process is not inﬂuenced by vapor depletion. To solve this problem, the strong dependence of nucleation rate on supersaturation is employed.

In the diffusion cloud chamber [6, 7], the nucleation and droplet growth processes are spatially separated. A diffusion cloud chamber consists of two parallel plates kept at different temperatures. A liquid layer of the substance under study is cov- ering the bottom plate. Because the plate is hot, the liquid evaporates and vapor driven by diffusion moves towards the upper cold plate. Along the vertical direction, the following supersaturation proﬁle exists: the supersaturation is increasing with height from the liquid surface, reaching a maximum where nucleation occurs and is then decreasing. The formed droplets are counted by optical means. The diffusion cloud chamber operates in a steady regime, when all heat and mass trans- port processes are settled. The conditions at which nucleation occurs are deduced from a solution of the energy and mass balance equations for the chamber.

A piston expansion cloud chamber [8] operates using the nucleation pulse principle [9] to separate nucleation and droplet growth in time. In the chamber the vapor is brought to a supersaturated state by a fast adiabatic gas expansion, caused by piston movement. After a short period of time, the piston slightly re-compresses the gas-vapor mixture, forcing nucleation to stop. Because the re-compression is adjusted in such a way that the gas-vapor mixture is weakly supersaturated, the formed droplets do not evaporate but grow further to macroscopic sizes. Such operational scheme allows to measure higher nucleation rates than in the diffusion chamber and also the determination of the nucleation conditions is appreciably simpliﬁed. The expansion wave tube [10, 11, 12] operates in a similar way as the piston expansion cloud chamber, and it also utilizes the nucleation pulse principle. In this tube, expansion and re-compression waves are created to realize the nucleation pulse principle (for details see Sec. 2.1). The absence of moving parts gives the possibility to measure higher nucleation rates and to operate at relatively high pressures.

6 1.3. APPLICATION

In the supersonic nozzle [13, 14], nucleation and droplet growth processes partially overlap. The gas-vapor mixture passes a converging-diverging nozzle, ex- pands isentropically and becomes supersaturated. The growing droplets are removed from the nucleation zone by the ﬂow, and are detected in the downstream part. In the supersonic nozzle, very high nucleation rates can be realized.

The experimental methods to determine nucleation rates are steadily improving. A large progress in the analysis of diffusion cloud chamber data was made by applying sophisticated 3D computational models. New methods are introduced in the experimental techniques: recently, Wyslouzil et al. [15] applied small angle neutron scattering (SANS) in their supersonic nozzle, which gives the possibility to probe the size and structure of clusters.

1.3 Application

Natural gas when it is produced from a well is a very complex mixture, the composition of which may vary in a wide range. Apart from methane, natural gas usually contains water, heavy hydrocarbons, carbon dioxide, hydrogen sulﬁde and other components. The contaminants in natural gas would cause problems in gas trans- portation and burning. Therefore, a careful separation of all unwanted condensable components is imperative. Also, there is a vast amount of natural gas reserves which have not yet been developed because they contain a high concentration of carbon dioxide: up to 0.2-0.3 molar fraction. The availability of a cheap and robust technology to purify natural gas with a high concentration of carbon dioxide will open these important energy resources for economical utilization.

Controlled droplet generation and droplet growth [16] is a very important technology for separation of methane from the other gases. This technology is based on the fact that most gases have a higher critical temperature than methane. By forcing their nucleation and droplet growth with a consequent removal of droplets, the natural gas can be puriﬁed.

In the traditional approach, water is removed from natural gas by means of adding glycol. Then, the gas-vapor mixture is brought into a supersaturated state by let- ting it pass a throttling valve, a typical isenthalpic process. Droplets are removed by means of coalescence, gravity separators, cyclones etc. In recent years alternative separation methods have been developed. An example is the Twister separa- tor [17], in which the gas is accelerated to a supersonic speed. Droplets are formed due to nucleation and condensation. The droplets are separated by generating a strong axial vortex in the supersonic ﬂow. A new technique is in development. It is based on isentropically cooling a gas-vapor mixture by means of an expansion turbine and then separating the droplets by means of a rotational particle separa-

7 CHAPTER 1. INTRODUCTION tor [18].

All processes have in common that homogeneous nucleation plays a key role in the formation of droplets and is determinate for mean droplet size and droplet number density. This thesis forms a continuation of research at TU/e, aimed at a quantitative and systematic study of nucleation rates in natural gas related mixtures in a wide window of thermodynamic state variables [19, 20, 21]. The present study focuses on nucleation in mixtures which contain one heavy alkane vapor (n-nonane) and two non-condensable components, methane and propane or carbon dioxide. It is expected that at relatively high pressures these non-condensable components will affect the nucleation process.

1.4 Thesis overview

Chapter 2 gives a detailed explanation of the experimental technique used for nucleation rate measurements: the nucleation pulse expansion wave tube. Special attention is paid to the accuracy of the pressure measurements. The Mie scattering techniques for droplet detection are critically discussed and their limitations are highlighted. A modiﬁcation of the mixture preparation setup is explained that allows to prepare ternary mixture with chosen compositions.

Chapter 3 deals with nucleation and droplet growth. First the kinetics of homogeneous nucleation is considered. An approach is suggested to reduce the multicomponent kinetics in gas-vapor mixtures to an effective unary one. Analytical formulas are derived for nucleation rates in quasi-unary multicomponent mixtures. The nucleation theorem formulated by Oxtoby and Kashchiev offers a powerful tool to deduce properties of critical clusters from nucleation rate data. The application of the nucleation theorem to binary mixtures is reformulated, such that it can directly be applied to determine the numbers of vapor molecules and carrier gas molecules in the critical cluster. In the last section, focused on droplet growth, it is explained how the diffusion coefﬁcient and phase equilibrium data can be deduced from droplet growth observations.

Chapter 4 deals with the gradient theory applied to n-nonane nucleation. The theory allows to estimate the energy of critical cluster formation – the important characteristic of the nucleation process. Also, the theory gives insight into the size dependence of cluster surface tension.

Chapter 5 reports n-nonane nucleation rate data for methane, methane/propane and methane/carbon dioxide carrier gas mixtures. For methane/n-nonane, this will be an extension of previous work, in which a strong inﬂuence of methane on n-nonane nucleation was already observed for pressures above 25 bar. Results for

8 REFERENCES the ternary mixtures are new. The critical cluster composition will be evaluated by means of the nucleation theorem. The inﬂuence of adding propane and carbon dioxide on n-nonane nucleation is addressed. The data obtained are compared with predictions from the quasi-unary theory.

References

[1] F.F. Abraham, Homogeneous Nucleation Theory, Academic Press, New York, 1974.

[2] D. Kashchiev, Nucleation; Basic Theory with Applications, Butterworth- Heinemann, Oxford, 2000.

[3] D. Kashchiev, J. Chem. Phys. 76, 5098 (1982).

[4] G.S. Springer, Advances in heat transfer, volume 14, chapter Homogeneous Nucleation, pages 281–345, Academic Press, 1978.

[5] R.H. Heist and H. He, J. Phys. Chem. Ref. Data 23, 781 (1994).

[6] J.L. Katz and B.J. Ostermeier, J. Chem. Phys. 47, 478 (1967).

[7] R.H. Heist and H.Reiss, J. Chem. Phys. 59, 665 (1973).

[8] P.E. Wagner and R. Strey, J. Phys. Chem. 85, 2694 (1981).

[9] E.F. Allard and J.L. Kassner, Jr., J. Chem. Phys. 42, 1401 (1965).

[10] F. Peters, Exp. Fluids 1, 143 (1983).

[11] K.N.H. Looijmans, P.C. Kriesels, and M.E.H. van Dongen, Exp. Fluids 15, 61 (1993).

[12] K.N.H. Looijmans and M.E.H. van Dongen, Exp. Fluids 23, 54 (1997).

[13] P.P. Wegener and A.A. Pouring, Phys. Fluids 7, 352 (1964).

[14] G. Lamanna, J. van Poppel, and M.E.H. van Dongen, Exp. Fluids 32, 381 (2002).

[15] B. E. Wyslouzil, G. Wilemski, and R. Strey, in Nucleation and Atmospheric Aerosols 2000: 15th International Conference, pages pp. 724–727, 2000.

[16] F.T. Okimoto and M. Betting, in Laurence Reid Gas Conditioning Conference, 2001.

9 CHAPTER 1. INTRODUCTION

[17] F. Okimoto and J. M. Brouwer, World Oil Magazine 223 (2002).

[18] J.J.H. Brouwers, R.J.E. van Wissen, and M. Golombok, Oil and Gas Journal (accepted for publication).

[19] K.N.H. Looijmans, Homogeneous nucleation and droplet growth in the coexistence region of n-alkane/methane mixtures at high pressures, PhD thesis, Eindhoven University of Technology, 1995.

[20] C.C.M. Luijten, Nucleation and Droplet Growth at High Pressure, PhD thesis, Eindhoven University of Technology, 1998.

[21] P. Peeters, Nucleation and condensation in gas-vapor mixtures of alkanes and water, PhD thesis, Eindhoven University of Technology, 2002.

10 Chapter 2

Wave tube experimental method

This chapter contains information on the experimental technique used to study nucleation phenomena in gas-vapor mixtures. The chapter begins with an introduction (Sec. 2.1) of the nucleation pulse method; the use of this method allows separating in time the nucleation and droplet growth processes which are taking place during the condensation of a gas-vapor mixture. The nucleation pulse method is implemented in the expansion wave tube. The gas-dynamic aspects of the tube design are discussed in Sec. 2.2. In the expansion wave tube a negative pressure pulse is formed in which nucleation occurs. The determination of pulse conditions is discussed in Sec. 2.3. The pulse pressure is measured with dynamic and static pressure transducers; a combined use of the transducers ensures a high experimental accuracy. The temperature is deduced from the pressure signal assuming an isentropic expansion. Also the duration of the nucleation pulse is extracted from the pressure signal. Errors in the determination of the nucleation conditions are evaluated. The droplets grown from the nuclei are detected by optical means, methods of detection being described in Sec. 2.4: 90◦-scattering and laser beam attenuation. The measured signals are interpreted with Mie scattering theory; this allows characterizing the size and number density of droplets in a droplet cloud. In Sec. 2.5 a description of the procedure for preparation of the gas-vapor mixture is given, together with a detailed explanation how a gas-vapor mixture with a chosen initial composition is obtained. A set of equations for the calculation of the mixture composition is presented, and the accuracy of mixture preparation is estimated as a function of the saturation conditions. Sec. 2.6 contains a detailed description of a typical experimental procedure. CHAPTER 2. WAVE TUBE EXPERIMENTAL METHOD

2.1 Nucleation pulse method

The nucleation pulse method is widely used in studying nucleation phenomena. The method allows to separate nucleation and droplet growth processes in time. Allard and Kassner [1] introduced the nucleation pulse method with an expansion- cloud chamber to investigate nucleation in helium/water gas-vapor mixtures. They stressed that the analysis of obtained data becomes much more accurate and reliable, because there is no need to study the combined complicated process of simultaneous nucleation and droplet growth.

Nucleation and droplet growth processes can be separated in space as well as in time. The ﬁrst, separation in space, is realized in diffusion-cloud chambers, where growing droplets are removed from a nucleation zone by gravity forces. The second, separation in time, is realized in expansion-cloud chambers and expansion wave tubes. In these facilities a short nucleation rate pulse is created. Nuclei of the new phase are formed during the pulse only (nucleation); after the pulse, these nuclei start to grow, becoming macroscopic droplets (droplet growth).

The nucleation pulse method is based on the fact that the nucleation rate strongly depends on supersaturation (see Fig. 1.3): a small change in supersaturation results in a large change in nucleation rate. Therefore, a pulse-shape supersaturation proﬁle gives a nucleation rate proﬁle with a sharp well-pronounced nucleation pulse (see Fig. 2.1). Most nuclei are formed during this nucleation pulse. After the pulse, the decrease of supersaturation quenches nucleation: the nucleation rate in the pulse is several orders of magnitude higher than after the pulse. The vapor still remains supersaturated after the pulse, S > 1, so that the formed nuclei grow to macroscopic sizes; such macroscopic droplets can be detected and counted. The duration of the nucleation pulse ∆t is measured; the number density of droplets n grown from the nuclei formed in the nucleation pulse is deduced by optical means; so, the nucleation rate is just the ratio:

n J = . (2.1) ∆t

The measured nucleation rate corresponds to the conditions (temperature, pressure, supersaturation) in the pulse, which are controlled experimentally. There- fore, with the nucleation pulse technique, nucleation-saturation J-S curves can be obtained at selectable constant temperatures and pressures.

There are several limitations of the nucleation pulse method. The pulse duration should be long enough so that steady-state nucleation is settled. The characteristic 6 time for establishing steady-state nucleation is estimated to be about 10− s for typical systems [2]. At the same time, the pulse duration should not be too long, because of vapor depletion. Vapor depletion occurs when an appreciable amount

12 2.2. PULSE EXPANSION WAVE TUBE

15 1020

12.5 1019

10 1018 ) -1 ∆t 17 s -3 S 7.5 10 J (m

5 1016

2.5 1015

6.5 7 7.5 8 8.5 9 9.5 10 10.5 11 t (ms)

Fig. 2.1: Supersaturation S (solid line) and nucleation rate J (dashed line) as functions of time t. In the pulse the supersaturation is two times higher than after the pulse. This leads to nucleation rates to be about four orders of magnitude larger in the pulse than after the pulse. During the nucleation pulse of duration ∆t new nuclei are formed. After the pulse the vapor is still supersaturated S ≈ 4, so that the nuclei can grow further. The supersaturation proﬁle is calculated with the RKS-EOS using the data from experiment 16dec03-01 on n- nonane nucleation in methane. The nucleation rate proﬁle is calculated with the quasi-unary nucleation theory. of vapor is consumed by growing nuclei; as a result, the conditions in the pulse will change. The characteristic time for vapor depletion is estimated in Appendix B for methane/water and methane/n-nonane gas-vapor mixtures.

In this study of nucleation phenomena an expansion wave tube is used. In this tube a gas-vapor mixture is brought in a supersaturated state by means of a fast isentropic expansion. The tube is designed such that the gas-vapor mixture expe- riences a pulse-shaped supersaturation proﬁle.

2.2 Pulse expansion wave tube

An expansion wave tube is a shock tube which is used in an unusual way. As a shock tube it has two sections separated by a diaphragm: the low pressure section (LPS) and the high pressure section (HPS). In shock tube literature these sections are called driver section and driven section [3], respectively. Normally, in a shock tube the properties of the initial shock wave moving to the end of the low pressure section is utilized to study physical and chemical processes in gases at high temperatures (see Fig. 2.2). But in an expansion wave tube, the initial expansion wave is utilized. The expansion wave is moving to and reﬂecting from the end of the high pressure section, thereby bringing the gas-vapor mixture in a supersaturated

13 CHAPTER 2. WAVE TUBE EXPERIMENTAL METHOD state. F. Peters ﬁrst designed and used an expansion wave tube [4] based on the nucleation pulse principle. To employ the nucleation pulse principle, in his tube a weak re-compressing wave was formed by partial reﬂection of the initial shock wave on a constriction in the low pressure section. The re-compression wave follows the expansion wave and they form together a negative pressure pulse, which results in the nucleation pulse in a supersaturated gas-vapor mixture.

Fig. 2.2: A typical shock tube and its corresponding space-time diagram. The shock tube has two sections separated by a diaphragm. After diaphragm rupture two waves appear – the initial expansion wave 1 moving towards the end of the high pressure section HPS and the initial shock wave 2 moving towards the low pressure section LPS. A contact discontinuity 3 separates the hot compressed gas from the cold expanded gas.

Peters’s original design was further developed and improved by Looijmans et al. [5, 6]. Their modiﬁed expansion wave tube has been designed such that a gas- vapor mixture is expanded in two steps: a fast step-like expansion is followed by a negative pressure pulse. In the modiﬁed expansion wave tube the nucleation pulse is formed by wave interactions with a local widening. The widening is a short part of the low pressure section with a diameter slightly larger than the rest of the tube. The widening leads to the formation of a well shaped supersaturation pulse (see Fig. 2.1). Furthermore, the duration of the supersaturation pulse and its magnitude can easily be varied by adapting the geometry of the the widening. An important advantage of Looijmans’s concept above the original design is that the wave interactions which happen after the nucleation pulse, only lead to weak pressure and temperature disturbances, which cannot cause any secondary after- pulse nucleation.

Prior to an experimental run, the high pressure section is ﬁlled with the gas-vapor mixture, and the low pressure section is ﬁlled with a carrier gas only. When the diaphragm is ruptured, the initial expansion wave and shock wave are formed; the expansion wave moves towards the end of the high pressure section and the shock wave moves towards the end of the low pressure section. Both waves are in-

14 2.2. PULSE EXPANSION WAVE TUBE dicated in Fig. 2.3 as 1 and 4, respectively. When the expansion wave reflects from the end-wall of the high pressure section, the local pressure and temperature in the measuring zone rapidly decrease towards a constant state. This constant state is maintained for a short period of time until the weak expansion wave arrives. The weak expansion wave is marked with 2 in Fig. 2.3. It is the result of a reflection of the initial shock wave from the first edge of the widening. The weak expansion wave causes a small decrease in pressure, which marks the beginning of the nucleation pulse. After reflection, the weak expansion wave leaves the measuring zone too. The pressure in the measuring zone is constant again until a weak re- compression wave arrives, 3 in Fig. 2.3. This re-compression wave is the result of a reflection of the initial shock wave from the second edge of the widening. The re- compression wave causes a slight increase of pressure which marks the end of the nucleation pulse: the negative pressure pulse d has been formed. After reflection of the re-compression wave, the pressure remains constant until the initial shock wave reflected from the end of the low pressure section arrives and destroys the supersaturated state. The experiment is terminated. This is a rather concise explanation of the principle of operation of the expansion tube and of the formation of a nucleation pulse. Looijmans [7] provides a more detailed description of the expansion wave tube gas-dynamics. Also, in Fig. 2.4 an experimental pressure recording is shown, and some additional information is given there.

Fig. 2.3: Formation of a pressure profile with a negative pulse in the modified expansion wave tube. The time origin is chosen at the moment of diaphragm rupture. The pressure is measured at the end-wall of the high pressure section. The most important waves formed in the tube are shown, denoted as 1 – the initial expansion wave; 2 – the weak expansion wave, caused by the shock wave reflected from the first edge of the widening; 3 – the weak re-compression wave, caused by the reflected shock wave from the second edge of the widening; 4 – the initial shock wave; and 5 – contact discontinuity. The pressure profile has a few distinctive parts: b – isentropic expansion of the gas-vapor mixture; d – pressure pulse (nucleation pulse); e – droplet growth stage.

The geometry of the expansion wave tube deﬁnes important characteristics for a nucleation experiment: the cooling rate, the duration of the supersaturation pulse, the depth of supersaturation, and the time duration of an experiment. The expansion wave tube used in this study has the following dimensions: the length of the high pressure section (HPS) is 125 cm, the length of the low pressure section (LPS)

15 CHAPTER 2. WAVE TUBE EXPERIMENTAL METHOD

60 P (bar) 50

a c b 30 0 5 10 15 20 25 30 35 40 45 t (ms)

Fig. 2.4: The pressure proﬁle measured in experiment 16dec03-01, methane/n-nonane mixture. The pressure is measured at 5 mm from the end-wall of the HPS. The negative pressure pulse is denoted with b. The disturbance a of the expansion wave is caused by non-ideal rupture of the diaphragm, and disturbance c is a result of interaction of the reﬂected expansion wave with the widening. is 923 cm, the inner diameter of the tube is 36 mm, the inner diameter of the widening is 41 mm, the length of the widening is 15 cm, the beginning of the widening is positioned 14 cm behind the diaphragm in the LPS, and the test zone is 5 mm from the end-wall of the HPS. Such geometry of the expansion wave tube with methane as a carrier gas gives a cooling rate of 5-15 K/ms, a ratio of supersaturations in and after the pulse of 2 4, a duration of the supersaturation pulse of − 0.30-0.55 ms, and a total test-time of 35-45 ms. A more detailed description of the expansion wave tube can be found in [7].

The high pressure section and the low pressure section are separated by a diaphragm. The design of this part of the expansion wave tube is critical for experiments. The diaphragm should open as fast as possible and parts of the ruptured diaphragm should not disturb the waves. The used diaphragms are made of poly- ester (polyethylene terephthalate). Their thickness depends on the pressure differences between the two sections just before the experiment. The thickness varies from 50 µm to 500 µm. The diaphragm is ruptured by heating it with a metal ribbon made of Kanthal. The metal ribbon is heated by a short electric pulse to a temperature up to 600 ◦C. The average time it takes to heat and to rupture a diaphragm is about 100 µs [7]. Loose parts of the ruptured diaphragm are pushed by the ﬂow behind the initial shock towards the end of the low pressure section, so that they do not disturb the formation of the pressure proﬁle. After the experimental run the remains of the diaphragm are removed from the low pressure section.

The measuring zone is located close to the end-wall of the high pressure section.

16 2.3. DETERMINATION OF PULSE CONDITIONS: PRESSURE, TEMPERATURE, TIME DURATION

This location of the measuring zone has several advantages: after the reﬂection of the expansion wave, the gas-vapor mixture is stagnant with respect to the tube walls which enables the observation of the development of a droplet cloud in the Lagrangian reference frame; it is easy to implement the optical setup for droplet cloud characterization.

2.3 Determination of pulse conditions: pressure, temperature, time duration

To analyze experimental data the conditions in the nucleation pulse should be known, i.e.: accurate values of temperature, pressure and pulse duration. Using these data the supersaturation of the vapor at pulse conditions can be inferred and the measured nucleation rate data can be correctly interpreted.

To measure the pressure we use two types of pressure transducers: quartz piezo- electrical pressure transducers and piezo-resistive ones. These transducers are suitable for different types of pressure measurements. Sometimes, the first type is called a dynamic pressure transducer; the second type is called a static pressure transducer. The quartz pressure transducer is applied to measure a fast changing pressure – it operates like a high-pass filter and the piezo-resistive transducer is applied to measure relatively slow-changing pressures or constant pressures – it operates like a low-pass filter. The frequency characteristics of the pressure transducers have some overlap, and this is used in the calibration procedure. In this study as a dynamic pressure transducer we used a Kistler type 603B transducer in combination with a Kistler 5001 charge amplifier, and as a static pressure transducer a Kistler 4073A50 for the pressure range 0-50 bar and a Kistler 4073A100 for 0- 100 bar. The signals from the transducers were recorded with a 100 kHz sampling rate by a Le Croy 6810 waveform recorder.

The dynamic and static pressure transducers have different calibration procedures. The static pressure transducer is calibrated prior to experimental runs, and the dynamic pressure transducer is calibrated in situ in each experiment with the signal obtained form the static pressure transducer. Both transducers are mounted near the end of the high pressure section at 5 mm distance from the end-wall. Dur- ing calibration of the static pressure transducer, close to its position a pressure calibrator is mounted. As a pressure calibrator we used a portable pressure calibrator Druck DPI 601. The calibrator pressure range was 0-100 bar – in this range the error of the calibration procedure is less than 0.075 bar.

During calibration, the high pressure section is ﬁlled stepwise with a carrier gas up to the highest pressure for a given set of experiments. Usually ten or more data

17 CHAPTER 2. WAVE TUBE EXPERIMENTAL METHOD

90 45 a Stat. PT 44 Dyn. PT 80 43

42 70 41

P (bar) 60 39 1.6 1.8 2 2.2 2.4 2.6

b 40

-5 0 5 10 15 20 25 30 35 t (ms)

Fig. 2.5: A comparison of signals from a static pressure transducer (solid line) and a dynamic pressure transducer (dashed line). In the pulse the transducer readings slightly differ: the static transducer gives higher values of pressure than the dynamic one. The pressure measured with the static transducer in a and b intervals is used to calibrate the dynamic transducer.

points are taken. Then, the pressure is decreased stepwise and about the same number of data points is taken. These data points of voltage versus pressure for the static transducer are linearly fitted. If the linearity of the fit is within the speci- fication for a given pressure transducer, then the calibration is accepted. Otherwise the calibration is rejected and the calibration procedure is repeated. The linearity is defined as the maximum deviation of the output of a device from a straight line fit of the calibration data [8]. For Kistler pressure transducers of type 4073A, the linearity is 0.3 percent of full scale, which limits the error of calibration for the 0-50 bar range to 0.15 bar and for the 0-100 bar range to 0.3 bar.

The dynamic pressure transducer is mounted at the same axial plane as the static one, and it is calibrated in situ, during an experimental run by comparing its signal with that of the already calibrated static transducer. To calibrate a dynamic transducer two specific intervals of the pressure history are used; in these intervals the pressure is approximately constant. The first is the time interval between diaphragm rupture and the arrival of the expansion wave to the end of high pressure section; the second is the time interval between the end of the nucleation pulse and the arrival of the reflected shock wave. The duration of both time intervals a and b is typically 3.5 ms, as is indicated in Fig. 2.5. The pressure levels during these constant states are recorded with the static pressure transducer. They are compared with the signal recorded by the dynamic pressure transducer and the characteristics of the dynamic transducer are calculated. Since the dynamic transducer behaves linearly to a high degree, one data point is sufficient for calibration.

18 2.3. DETERMINATION OF PULSE CONDITIONS: PRESSURE, TEMPERATURE, TIME DURATION

The natural frequency of the static transducer is more than 110 kHz, the natural frequency of the dynamic transducer is more than 400 kHz. Since the typical time scale for the duration of the nucleation pulse is 0.5 ms, both transducers should give the correct pulse pressure level. Fig. 2.5 shows a detailed comparison of both signals in the nucleation pulse. The observed systematic difference between the signals is 0.2 bar for a total pressure range of 40 bar, and it cannot be explained. Furthermore, the difference is of the same order of magnitude as the experimental error of the individual transducers. In the evaluation of the experiments we took the dynamic pressure data as a basis.

A remark should be made with respect to the sensitivity of the 603B quartz transducer to the surface heat flux. It is well know that such a heat flux induces a temperature gradient inside the transducer which could lead to erroneous interpretation of the signals. The effect is suppressed by adding a protection layer of silicon rubber on top of the transducer with a thickness of 0.3 mm. In addition, when pressure remains constant, the signal from the dynamic transducer can slightly change, because the quartz crystal is not fully electrically isolated and it is losing its charge in time. But since the RC-time of the dynamic pressure transducer and the charge amplifier far exceeds the test time, the losing of charge does not influence the experimental accuracy. The RC-time can be set by changing the corresponding modes of the charge amplifier. In all our experiments the charge amplifier was set into the “long” mode, corresponding to a RC-time of the order of 10 s.

The nucleation pressure is defined as the average pressure over the pulse duration. An example of the negative pressure pulse is shown in Fig. 2.6. The time position of the pulse end is very well defined, it is indicated with t2. The time that marks the pulse beginning is somewhat ambiguous, it is indicated with t1. The pulse duration is the difference between these two times: ∆t = t t . The time t 2 − 1 1 is chosen visually, but in a such way that the pressure difference between the pressure at t1 and the average pressure is approximately the same as the pressure fluctuations in the middle region of a pressure pulse. The uncertainty in time t1 is less than 0.1 times the pulse duration ∆t, which gives a relative error of 0.1 in the measured nucleation rate. The pressure fluctuations ∆P in the pulse (see Fig. 2.6) are of the same order as the error in the pressure measurements. As was demonstrated by Luijten [9], a choice of slightly different t1 causes a shift of a (J, S) point along the nucleation isotherm, so the slope of the nucleation isotherm is insensitive to a variation of t1.

It is not possible to measure the gas-vapor temperature in the nucleation pulse directly, so we evaluate the temperature from the pulse pressure. The gas-vapor mixture is expanding isentropically, so that temperature and pressure are uniquely related. We do not take into account the inﬂuence of the vapor on the properties of the gas-vapor mixture, because the vapor concentration is very small – it varies be-

19 CHAPTER 2. WAVE TUBE EXPERIMENTAL METHOD

35 P (bar)

34 ∆P

33 ∆t t1 t2

32 6.75 7 7.25 7.5 7.75 8 t (ms)

Fig. 2.6: A negative pressure pulse. The pulse duration ∆t is deﬁned as ∆t = t2 − t1. Both t1 and t2 are determined visually. The pressure ﬂuctuations ∆P in the pulse are of the same order as the error in the pressure measurements.

tween 50-300 ppm. Thus, the relative contribution of the n-nonane vapor in the molar specific heat of the mixture is insignificant; to see this, the ratio of partial molar heats should be estimated. At the temperature range of 200-300 K and at 40 bar, the molar specific heat of n-nonane varies between 219.2-282.4J/(mol K); for methane, which is the main carrier gas component, the molar specific heat varies between 59.3-40.3J/(mol K). Then, at the highest concentration of n-nonane in the mixture, the ratio of the partial molar heat capacities C y /(C y ) is of the order p,c1 · c1 p,c9 · c9 of 3 102; obviously the n-nonane almost does not contribute in the molar specific × heat of the mixture. For example, for an isentropic expansion with a temperature drop of about 60 K, not taking the contribution of n-nonane into account causes an error of 0.06 K in the temperature calculations. The real gas properties of the carrier gas mixtures are taken into account. These real gas properties are evaluated on the basis of equations of state with mixing rules.

To evaluate the relation of pressure and temperature in an isentropic expansion of the gas mixture, Maxwell’s relations are combined with the ﬁrst law of thermodynamics. Then, it follows that

dT R Z T ∂Z dP = a 1 + , (2.2) T C Z ∂T P p P where T and P are the mixture temperature and pressure, Z is the compressibility factor of the mixture, Cp is the molar speciﬁc heat of the mixture, and Ra is the universal gas constant. The compressibility Z and the speciﬁc heat Cp weakly depend on temperature. So, equation (2.2) can be solved for a sequence of small

20 2.3. DETERMINATION OF PULSE CONDITIONS: PRESSURE, TEMPERATURE, TIME DURATION

pressure steps from Pi to Pi+1:

T P αi i+1 = i+1 , (2.3) T P i i where Ra Z(Ti,Pi) Ti ∂Z αi = 1 + . (2.4) Cp (Ti,Pi) Z(Ti,Pi) ∂T Pi ! This gives a discrete scheme to integrate equation (2.2) – starting from the initial pressure P0 and temperature T0. The pressure P0 is the pressure recorded by the static pressure transducer before the expansion wave reaches the end of the high pressure section. The temperature T0 is the temperature of the high pressure section which is measured just before diaphragm rupture. The temperature is measured with an Omega K-type thermocouple together with an Omega type 871A digital temperature detector. The error of the temperature measurements is less than 0.1 ◦C. A series of calculations is performed, so that the pressure step of the discrete scheme is decreased stepwise until the temperature varies less than 0.1 ◦C between two successive calculations.

The molar speciﬁc heat of the mixture, Cp in (2.2) is calculated using an equation of state. The heat capacity of a real gas mixture can be split in two parts [10]:

id Cp = Cp + ∆Cp, (2.5)

id where Cp is the specific heat of mixture in the ideal-gas state and ∆Cp is the residual specific heat. For the mixture in the ideal-gas state, the specific heat is determined by id id Cp = yiCp,i, (2.6) i X id where yi is the molar fraction of the i-th gas. The molar specific heats Cp,i of the i-th pure gas in the ideal-gas state are known functions of temperature for methane, propane, and carbon dioxide [10]. The residual molar specific heat ∆Cp is found from V 2 2 ∂ P T (∂P/∂T )V ∆Cp = T 2 dV Ra. (2.7) ∂T V − (∂P/∂V )T − Z∞ As soon as the value of the specific heat is known for given Ti and Pi, it is inserted in (2.4) and the next step in the discrete scheme is performed.

During an experimental run the pressure is recorded. Then, the nucleation temperature, the supersaturation, and the pulse duration are deduced from the obtained pressure proﬁles. The last quantity to be determined is the number density of droplets born in the nucleation pulse, at this known pressure and temperature.

21 CHAPTER 2. WAVE TUBE EXPERIMENTAL METHOD

2.4 Detection of macroscopic droplets

The characteristics of a droplet cloud are determined by optical means: light scattered by the droplets transfers information about the number density of droplets and their sizes. When a monochromatic light beam illuminates a cloud of dielec- tric particles, it will be scattered or/and absorbed. The net result of absorption and scattering is called extinction (extinction = scattering + absorption). If the particles do not absorb the incident light and the wavelength of scattered light is the same as the wavelength of incident light, the scattering is deﬁned as elastic. The theory of elastic scattering by particles is very well developed [11, 12, 13].

The problem of describing the elastic scattering on particles can be split in two steps. The ﬁrst step is to focus on the scattering process from a single particle, and to ﬁnd out the scattered light properties as a function of the incident light properties, particle size, particle composition, and particle shape. The second step is to investigate cooperative effects of scattering by a cloud of particles.

The typical scattering geometry for a single particle in the laboratory frame is shown in Fig.2.7. An incident light beam with intensity I0 illuminates a particle. The light propagates along the X direction, as is indicated by the wave vector −→k ( −→k = 2π/λ). The light has polarization −→E which is normal to the scatter- ⊥ ing plane. The scattering plane is formed by the direction of propagation of the incident light and by the direction in which scattering is observed. A particular direction of scattering is characterized by two angles θ and φ. The scattering angle θ is the angle between the directions of propagation of the scattered and incident light. The azimuthal angle φ is the angle between the scattering plane and the plane of polarization. A special case is when the light is scattered in forward direction – the light passes through the particle. The light scattered forward interferes with the incident light wave and causes the attenuation of the transmitted light beam.

The scattering by a single particle is described by the Maxwell equations with appropriate boundary conditions. The Mie theory gives an analytical far-ﬁeld solution of the Maxwell equations for scattering by a spherical particle of arbitrary radius rd. Furthermore, the particle should be homogeneous, isotropic, and non- magnetic. The derivation of the solution can be found elsewhere [12]. The solution provides an expression for the scattering intensity Isc in a solid angle (dθ, dφ) and for the geometry as in Fig. 2.7:

Isc (m,θ,φ,α) = I 0 S (α,θ,m) 2 sin2(φ) + S (α,θ,m) 2 cos2(φ) sin θdθdφ, (2.8) 2 2 | 1 | | 2 | 2 −→k −→d Z

22 2.4. DETECTION OF MACROSCOPIC DROPLETS

Fig. 2.7: Scattering by a single particle in the laboratory frame. The scattering angle θ is the angle between directions of propagation of the scattered and incident light. The azimuthal angle φ is the angle between the scattering plane and the plane of polarization.

where −→d is the distance from the scattering particle to a detector, m is the relative index of refraction of the particle substance, and α is the size parameter. The size parameter is a dimensionless quantity which characterizes the particle size relatively to the wavelength of scattered light

2πr α = d = −→k r . (2.9) λ d

The complex amplitude functions S1 and S2 in (2.8) are speciﬁed by the Mie theory, and have an oscillatory character as functions of α. The exact expressions of the amplitude functions can be found in [11].

Knowledge on how light is scattered by a single particle is not enough to describe the scattering by a cloud of particles. If the particles are sufﬁciently far from each other, then the particles scatter light independently – there is no systematic relation between the phases of light scattered by different particles. In practice it means that the scattered light intensity can be obtained by adding the intensities of light scattered by different particles without taking into account phase relations. To ensure independent scattering, the mutual distance between particles should exceed 3 times the particle radius [11]. In our experiments this condition is always fulﬁlled; since even in the worst case, when the number density and 14 3 the droplet size have the highest observable values, n = 0.5 10 m− and max × 7 r = 8.2 10− m , the relative mutual distance between particles is d,max × 1/3 4 − πn r3 10. 3 max d,max ≈ 23 CHAPTER 2. WAVE TUBE EXPERIMENTAL METHOD

If particles in the cloud are close to each other, each particle will be exposed to light scattered by the other particles. Such process is called multiple scattering. Swan- son et al. [14] studied theoretically and experimentally limits of application of the Lambert-Beer law for transmission light measurements. They found that multiple scattering is certainly negligible when the optical depth βl (2.10) is less than unity. They claimed that the Lambert-Beer law is valid up to an optical depth βl = 10. And they claimed that the use of the Lambert-Beer law should be restricted when the optical depth exceeds ten. For all our experiments these conditions are fulﬁlled (see Fig. 2.8), and the Lambert-Beer law can be safely applied for interpretation of transmission light measurements.

103

α1 α3 α5 102

101

100 l β

10-1

10-2 Optical depth 10-3 10 -4 n = 10 10 n = 1011 n = 1012 13 10-5 n = 10 n = 1014 n = 1015 10-6 1 2 3 4 5 10 15 20 Size parameter α

Fig. 2.8: Optical depth in dependence of size parameter and droplet number density 3 n (1/m ) for the present experimental setup. When the optical depth βl is less than 10 (thick dashed line) multiscattering is negligible. At least one Mie peak α1 should be observed in an experimental run to determine the droplet number density. For all experiments, the Lambert-Beer law can be used to interpreted transmitted light intensity signals.

The optical setup has been arranged in such a way that the transmitted light intensity and the scattered light intensities are measured as functions of time. During this project the optical setup was improved. A new laser with better characteristics was installed. The use of the new laser allowed us to simplify the original optical scheme shown in Fig. 2.9. A laser beam from an argon-ion laser illuminates a cloud of droplets in the measuring zone. At the laser output, the light beam is composed of two wavelengths: 488.0 nm and 514.2 nm. The 488.0 nm light is removed from the beam by a filter F. After the filter, the 514.2 nm laser beam passes a polarizer P, adjusting the beam polarization in the direction normal to the scattering plane. With a semipermeable mirror M the laser beam is split in two parts. The deflected part of the beam proceeds to a photodiode PD1. The deflected beam is measured by a Telefunken type BPW-34 photodiode. This is the reference optical signal which is used to eliminate the laser own noise from the scattering signals. The light beam enters the high pressure section through a side-wall glass window. The center of

24 2.4. DETECTION OF MACROSCOPIC DROPLETS the window is positioned at a distance of 5 mm from the end-wall. The light beam is partly reflected on the window surfaces. To avoid interference with the reflected beams, the window is slightly inclined, so that the reflected beams cannot reach the measuring zone. The light beam passes the measuring zone and illuminates the cloud of droplets. The light scattered at an angle of 90◦ leaves the measuring zone through the end-wall window. The scattered light is collected with lens L1 and diaphragm D1 and it is focused on the photomultiplier PM (Hamamatsu type

1P28A). The lens and diaphragm collect light within a solid angle θ = 90 1.15◦ ± and φ = 90 6◦. The transmitted light leaves the measuring zone through a side- ± wall window. It passes diaphragm D2, which blocks the light scattered at small angles. The transmitted light is focused on the photodiode PD2 (Telefunken BPW 34) by the lenses L2 and L3. Before the transmitted light beam reaches the photodiode, its intensity is decreased by the multiple reﬂection on glass plate GP. The optical geometry realized in this setup is similar to that of Fig. 2.7: the incident light polarized normally to the scattering plane, the transmitted light, and the 90◦-scattered light. The reference signal, scattered signal, and transmitted signal are recorded by the Le Croy recorder with a 100 kHz sampling rate. The recording of optical signals is synchronized with the recording of pressure signals. So, at any given moment of time the pressure, transmitted and scattered light intensities are known.

Fig. 2.9: The original optical setup. The two-wavelength light beam from the argon-ion laser AIL passes the ﬁlter F which leaves only the 514.2 nm component in the beam. Polarizer P adjusts the plane of light polarization in the direction normal to the scattering plane. The ◦ reference signal is recorded with photodiode PD1. The 90 scattering signal is recorded with a photomultiplier PM. After its intensity is decreased by multiple reﬂections on a glass plate GP, the transmitted signal is recorded with photodiode PD2.

25 CHAPTER 2. WAVE TUBE EXPERIMENTAL METHOD

The new optical scheme is shown in Fig. 2.10. A 532.0 nm laser beam illuminates the cloud of droplets, it is produced by the diode laser LasNova type GL 3220 TO1. The laser has an internal polarizer which can be adjusted, so that at the exit the light is polarized in the direction normal to the scattering plane. The new laser is more stable than the old one; therefore, there is no need to split the incident light beam and to measure a reference signal. As a result the input cascade of the optical scheme can be simpliﬁed. The rest of the optical setup is the same as the original one.

Fig. 2.10: The new optical setup, in which a diode laser DL is installed. The laser has an excellent signal-to-noise ratio and polarization characteristics; because of these advantages of the new laser over the old one, the optical scheme is simpliﬁed in comparison with the one in Fig. 2.9.

The information about the size of the droplets is extracted from the scattering signal. The method is illustrated by Fig. 2.11, where two intensity signals are shown: the 90◦-scattering intensity for a single particle predicted by Mie theory (2.8) and the 90◦-scattering intensity observed in an experiment for the droplet cloud. The Mie scattering intensity is plotted as a function of size parameter α (Fig. 2.11(a)). The intensity signal oscillates – it has a distinct maxima-minima pattern. In its turn, the scattering signal observed in an experiment is plotted as a function of the square root of time t (Fig. 2.11(b)). The square root of time is used because the droplet growth is diffusion-controlled, and the size of the droplets is proportional to the square root of time, r √t (see Sec. 3.4). The intensity signal also d ∝ oscillates, and it also has a distinct maximum-minimum pattern. It is possible to identify maxima and minima in one ﬁgure that correspond to maxima and min-

26 2.4. DETECTION OF MACROSCOPIC DROPLETS ima in the other. In Fig. 2.11 the corresponding maxima and minima are marked with the same numbers. Because of the correspondence between the theoretical Mie scattering pattern and the observed scattering, a distinct size can be ascribed to the droplets at a distinct time, and a droplet growth curve can be derived. The striking similarity between the scattering intensity caused by a single particle and the scattering intensity caused by a droplet cloud also means that the droplet cloud has an almost mono-dispersed size distribution. All droplets have approximately the same size, which gives an experimental proof of the correct implementation of the nucleation pulse principle.

When the scattering signal has a few extrema it is sometimes difﬁcult only by visual inspection to relate them with the Mie extrema. Especially in the case of a weak scattering signal, there is a possibility that the ﬁrst maxima and minima are overlooked because of noise. To solve the problem, a reasonable guess should be made about the extrema numeration. After comparison with the Mie extrema the droplet growth curve is plotted and the curve is extrapolated towards the nucleation pulse. If the curve is close to the nucleation pulse at rd = 0 m, then the guessed numeration is the correct one (see Fig. 2.12), otherwise the numeration should be shifted and the procedure should be repeated.

The information on the number density of droplets in the cloud is extracted from the transmitted intensity signal. Mie scattering theory yields the transmitted light intensity for a single particle in the far-ﬁeld zone. Knowing the transmitted intensity from a single particle, the transmitted intensity for a homogeneous cloud of droplets can be derived in the form of the Lambert-Beer law

Itr 2 = exp πrdnQext (m, α) l = exp ( βl) , (2.10) I0 − − where l is the length of the optical path through the droplet cloud, Qext is the extinction efficiency, β is the extinction coefficient, and βl is the optical depth. Equa- tion (2.10) is valid if multiple scattering is negligible. The exact expression for the extinction efficiency Qext from Mie theory is given in [11]. Since, the size of the droplets rd is known from the scattering signal the only unknown in (2.10) is the droplet number density n. By solving equation (2.10) for n the number density of the droplets is found.

In the experiments the vapor always condenses in the presence of a carrier gas; as a result, the growing droplets consist of liquid mixture of the vapor and the carrier gas molecules. Obviously, such liquid mixture has a refractive index different from that of the pure liquid. For different refractive indices the Mie theory predicts scattering intensities and extinction efﬁciencies, which are not the same.

The dependence of the refraction index on the composition of the liquid mixture

27 CHAPTER 2. WAVE TUBE EXPERIMENTAL METHOD

6 5

(a.u) 3 sc,mie I 3 2 6

1 4 1 2 0 0 1 2 3 4 5 6 7 α (a)

0.7 5 0.65 6 0.6

0.55

0.5

(volt) 3 0.45 sc,exp

I 4 0.4

0.35 1

0.3 2

0.25 -10 -5 0 5 10 15 20 25 30 35 40 t1/2 (ms)1/2 (b)

Fig. 2.11: The Mie scattering intensity signal Isc,mie as a function of size parameter α, and the measured scattering intensity signal Isc,exp as a function of the square root of time t. It is easy to recognize the corresponding extrema in the ﬁgures: they are denoted with the same numbers. Therefore, the size parameters of the droplets are known at these extrema, and a droplet growth curve can be derived (see Fig. 2.12). The shown data are taken from the 16dec03-01 methane/n-nonane experimental run.

28 2.4. DETECTION OF MACROSCOPIC DROPLETS

80 0.3

6 0.25 70 5 0.2 60 2 m)

0.15 µ ( 2 P (bar) d

4 r 50 3 0.1

40 2 0.05 1

30 0 0 5 10 15 20 25 30 35 40 t (ms)

Fig. 2.12: A pressure proﬁle together with a n-nonane droplet growth curve. The droplet growth data are obtained from the scattered and transmitted signals. The numeration of the data points is the same as in Fig. 2.11. Because the droplet growth starts at the nucleation pulse, an extrapolation of growth curve to zero size should cross the time axis close to the nucleation pulse position if the corresponding extrema of the Mie and experimental scattered intensities are correctly interpreted.

can be evaluated with the Lorentz-Lorenz formula:

m2 1 4π − = N ρ a , (2.11) m2 + 2 3 a mix mix where ρmix is the mixture molar density, amix is the mixture polarizability, and

Na is the Avogadro constant. Assuming a linear additive rule for the polarizabilities of the pure components of the mixture [15], the mixture polarizability can be expressed as

amix = xiai, (2.12) i=1 X where xi is the molar fraction of the i-th component The polarizabilities of methane, propane, carbon dioxide, and n-nonane are known, and they are listed in [16]; the molar density ρmix of a mixture can be calculated with an appropriate equation of state.

The calculation of refractive indices of liquid mixtures are performed at a typical experimental temperature of 240 K, and at the highest experimentally achievable pressure of 40 bar. The higher the pressure, the more carrier gas is dissolved in the liquid and the higher the liquid mixture molar density. To evaluate the mixture compositions and molar densities at the given temperature and pressure, vapor-liquid equilibrium ﬂash calculations are carried out with the RKS-EOS for the initial mixture compositions. The initial mixture compositions taken are typical experimental ones. As the refractive index of the environment the refractive

29 CHAPTER 2. WAVE TUBE EXPERIMENTAL METHOD index of the equilibrium gas-phase mixture is taken, which is also calculated with (2.11).

The calculations reveal that the relative refractive indices for different mixtures are about the same (see Tab 2.1); the only exception is the C1/C3(0.03)/C9 mixture. This is because n-nonane mainly contributes to the polarizability of the liquid phase: it has the highest polarizability and the highest molar concentration; moreover for different mixtures the change of the liquid mixture molar density is too small to inﬂuence the refractive index signiﬁcantly. In addition, the relative refractive index of the mixture is close to that of pure methane, because the molar densities of the liquid and gas phases increase as well as the refractive indices of both phases, keeping the relative index almost constant. In the calculations we used a relative refractive index of 1.405. The difference between the used index and the real one is less than 0.02, such difference results in a relative error less than 0.02 in determining the droplet number density with (2.10).

initia mixture m pure n-nonane 1.405 C1-C9 1.400 C1-C3(0.010)-C9 1.391 C1-C3(0.030)-C9 1.365 C1-CO2(0.035)-C9 1.400

Tab. 2.1: Relative indices m of refraction for initial mixtures at 240 K and 40 bar. In the parentheses the molar fraction of the corresponding component is indicated.

The relative refractive index of the initial mixture with a 0.03 molar fraction of propane is noticeably smaller than the index of pure n-nonane. The reason for that is the high molar fraction of propane in the liquid phase: the fraction is 0.31, which is comparable with the molar fraction of n-nonane 0.41. The presence of propane in the liquid phase in such high concentrations appreciably inﬂuences the mixture polarizability and even more the mixture molar density.

The use of the pure n-nonane refractive index for the C1/C3(0.03)/C9 mixture can cause a relative error in the determination of the droplet number densities as high as 0.3 (see Fig. 2.13); in fact the situation is even worse because the relative error in the droplet number density

δn Q (α ,m ) Q (α ,m ) ext c9 c9 − ext mix mix n ≈ Qext(αc9,mc9) also depends on the size parameters αc9, and αmix which follow from the scattered intensity signals for the corresponding extrema. The latter have different size parameters for different refractive indices, and the difference is increasing with droplet size. In 40 bar experiments only a few Mie scattering extrema are observed, so the droplet number densities are determined from the transmitted intensity signals for droplets having α-values in the range of 2-5. In that range for the droplet number

30 2.5. MIXTURE PREPARATION AND INITIAL MIXTURE COMPOSITIONS density the relative error can be up to 0.2: to eliminate this error the relative refractive index predicted by (2.11) has been used in the analysis of experiments in mixtures containing propane.

0.3 m = 1.391 m = 1.365 0.2

0.1 ext,c9

)/Q 0 ext,c9

- Q -0.1 ext,mix

(Q -0.2

-0.3

-0.4 0 1 2 3 4 5 6 7 8 9 10 α

Fig. 2.13: Relative difference between the extinction efﬁciencies for a liquid droplet consisting of n-nonane/propane and for a pure n-nonane droplet. The dashed line is the relative difference for a C1/C3(0.03)/C9 initial mixture, and the solid line – for a C1/C3(0.01)/C9 initial mixture.

Some of the calculations of Mie scattered intensities and transmitted intensities presented here were performed with the program1 MiePlot by Laven [17]. In the program the classic BHMIE algorithm [13] is implemented.

2.5 Mixture preparation and initial mixture compositions

It is crucial for nucleation experiments to prepare a gas-vapor mixture with pre- cisely known composition. The nucleation rate strongly depends on supersaturation, which is proportional to the vapor molar fraction (A.15), which varies in the range of 50-300 ppm. To prepare a mixture with such a low vapor content the saturation section designed by Hrubý was used. A more detailed description than given here can be found in his report [18].

The gas-vapor mixture is prepared in the saturation section of the setup. The section can be divided in four main parts: a gas supply part, a ﬂow control part, saturators, and a static mixer. The gas supply part is a set of gas cylinders connected with pressure transducers to the ﬂow control part. The gas cylinders can

1Programs and codes for different scattering situations (coated spheres, multispheres, scattering on spheroids, slab scattering) can be downloaded from http://atol.ucsd.edu/scatlib.

31 CHAPTER 2. WAVE TUBE EXPERIMENTAL METHOD contain the dry carrier gases – methane, carbon dioxide, or carrier gas mixtures – methane/propane, methane/carbon dioxide. High purity gases and gas mixtures are used in the experiments. The gases are analyzed by the Hoekloos gas delivery company; the presence of water in the gases is less than 5 ppm and the presence of hydrocarbon impurities is less than 2 ppm.

The flow control part consists of a set of mass flow controllers, a flow control unit, and a manifold assembly. The flows through the mass flow controllers are regu- lated with the flow control unit Brooks 0154. The flow control unit is combined with a PID-controller; the controller reads a signal from a pressure transducer, which measures the pressure in the saturators (PT in Fig. 2.14). The pressure signal from the transducer is compared with a PID-controller setting. The flow control unit ad- just the flows, such that the transducer signal and the setting of the PID-controller are equal, while keeping the ratios of the different flows constant.

The used Brooks 5850S mass ﬂow controllers have different operational ranges

3000, 1500, and 300 sccm N2. If the carrier gas is methane, then the corresponding operational ranges are 1.2 times lower. The manifold assembly allows to con- nect the most suitable flow controllers for given flow settings, and therefore to increase the accuracy of the mixture preparation. In addition, by means of the manifold assembly, the dry diluted flows and the flow through the saturators are arranged.

In the saturators, n-nonane vapor is added to the dry carrier gas. The schematic drawing of the saturator part is shown in Fig. 2.14. The carrier gas flow Qc9 bub- bles through the two saturators S1 and S2 connected in series. The saturator is a stainless steel vessel completely filled with three layers of glass beads: the diam- eters of the beads are 2 mm, 4 mm, and 6 mm. The bead layers spread the carrier gas in liquid n-nonane improving the quality of saturation. The saturators are half filled with n-nonane. The purity of the n-nonane is better than 99 %. The saturators are submerged in a thermostatic bath TB, so that the temperature in the saturators is held constant. The saturator temperature is always a few degrees lower than room temperature to prevent the condensation in the saturated gas flow after the exit from the saturators. The temperature in the saturators Tsat is measured with platinum-resistor Tempcontrol thermometers T1 and T2. The error of the temperature measurements is within 0.02 K. The pressure in the saturators P sat is measured with a pressure transducer PT of Druck type PMP 4070. A set of pressure transducers with different ranges 35 bar, 70 bar, and 135 bar has been used. The pressure in the saturators is known prior to experimental runs, so it is possible to choose the most appropriate pressure transducer. The error of the pressure mea- 4 surements is less than 8 10− of full scale of the transducer. After the saturated × gas flow Qc′ 9 has passed the saturators, it is diluted with the dry carrier gas flows

Qc1 and Qc3. The obtained gas-vapor mixture then passes the static mixer, which eliminates any possible ﬂuctuations of the mixture composition.

32 2.5. MIXTURE PREPARATION AND INITIAL MIXTURE COMPOSITIONS

Fig. 2.14: The saturator part of the setup. The dry methane flow Qc9 passes two saturators ′ S1 and S2, which are half filled with liquid n-nonane. The output gas-vapor flow Qc9 is diluted with the dry gas flows Qc1 and Qc3 to prepare a ternary methane/propane/n-nonane mixture. The temperature in the saturators is measured with platinum-resistor thermometers T1 and T2, and the pressure is measured with transducer PT. To keep the saturators at constant and known temperature, they are submerged into thermostatic bath TB.

33 CHAPTER 2. WAVE TUBE EXPERIMENTAL METHOD

It is possible that for some reason the gas flow will not be completely saturated. In order to check whether the gas flow is saturated, the following method has been applied. The mixture from the saturators is analyzed with a gas-chromatograph. The same analysis is performed for mixture which is obtained when the gas flows only through the first saturator, S1 in Fig. 2.14. To bypass the second saturator the valve V should be opened. If the gas-chromatographic analysis for both mixtures give the same results, then we are sure that the gas is saturated after having passed one saturator.

For known thermodynamic conditions and molar ﬂows through the saturation section, the calculation of the mixture composition of the output molar ﬂow Qt is straightforward. The molar fractions y ,y ,y of gases in the mixture are { c9 c1 c3} given by the following set of equations:

yeq Q y = c9 c9 , (2.13) c9 1 yeq Q − c9 t Qc3 yc3 = Yc3 , (2.14) Qt y = 1 y y , (2.15) c1 − c9 − c3 P (T ) yeq = f T ,P sat s,c9 sat , (2.16) c9 e sat P sat Q c9 Q = + Q + Q ; (2.17) t 1 yeq c1 c3 − c9 where yc9 is the vapor (n-nonane) molar fraction, yc1 is the molar fraction of the

ﬁrst carrier gas (methane), yc3 is the molar fraction of propane, Ps,c9 is the saturation vapor pressure, and fe is the vapor enhancement factor. The second dry molar

ﬂow Qc3 is a mixture methane/propane, and Yc3 is the molar fraction of propane in that mixture. If the second carrier gas is a methane/carbon dioxide mixture, the subscript c3 should be changed to co2 in all equations. The meaning of the variables remains the same.

The used mass flow controllers were calibrated with a Brooks 1067 Vol-U-Meter gas-calibrator prior to experimental runs. The calibration of a controller is performed for each carrier gas used in the experiments. The reading from the controllers slightly depends on the inlet pressure and temperature of a carrier gas. During the calibration, the inlet pressure and temperature were close to those in the experiments. For every mass flow controller the calibration points were taken 3 in the full operational range. The error in the flow measurements is less than 10− of the full range of a mass flow controller. Some difficulty has been encountered during the calibration of flow controllers with pure carbon dioxide. When the pressure drop over the mass flow controller was higher than 15 bar the carbon dioxide flow freezes the controller because of Joule-Kelvin “cooling”. To eliminate the freezing the carbon dioxide flow was preheated up to 40 ◦C and the pressure drop was kept less than 5 bar. In the experiments the preheating temperature and the

34 2.5. MIXTURE PREPARATION AND INITIAL MIXTURE COMPOSITIONS pressure drop were the same as during the calibration. This complication of the calibration procedure leads to a decrease in the accuracy of calibration. For pure carbon dioxide the error was less than 0.01 of the full range of the mass ﬂow controller.

The error in the composition of the prepared mixture depends on the accuracy of flow measurements and on the accuracy of determining the equilibrium composition in the saturators. To estimate the contribution of these factors to the uncertainty of the n-nonane molar fraction, the general expression for error propagation ∂y 2 ∂y 2 ∂y 2 δy = δx + δx + . . . + δx (2.18) ∂x 1 ∂x 2 ∂x n s 1 2 n is applied to equation (2.13). But first, (2.13) is somewhat simplified: the molar fraction y is very small, so 1 y is substituted with 1; and, the sum of carrier c9 − c9 gas flows Qc1 + Qc3 is substituted with a total carrier gas flow Qc1,c3. This yields expression eq Qc9 yc9 = yc9 (2.19) Qc1,c3 + Qc9 which can be easily analyzed. The relative error in yc9 because of the uncertainty in the flow through the saturator is

1 ∂y Q δQ ǫ (y Q ) = c9 δQ = 1 c9 c9 . (2.20) c9| c9 y ∂Q c9 − Q + Q Q c9 c9 c9 c1,c3 c9

The flow through the saturators is always more than 0.1 of the full flow range of 3 the mass flow controller and the uncertainty in the flow is 2 10− of the full range × 3 plus 7 10− of the actual flow, so ǫ (y Q ) < 0.03. Similarly, the relative error × c9| c9 in yc9 because of the uncertainty in Qc1,c3 is

1 ∂y δQ ǫ (y Q ) = c9 δQ = c1,c3 , (2.21) c9| c1,c3 y ∂Q c1,c3 Q + Q c9 c1,c3 c1,c3 c9

with the same result ǫ (yc9 Qc1,c3) < 0.03. Finally, the contribution of the uncer- eq | tainty in yc9 is 1 ∂y δyeq ǫ (y yeq) = c9 δyeq = c9 , (2.22) c9| c9 y ∂yeq c9 yeq c9 c9 c9

eq sat where the δyc9 is deﬁned by the uncertainties in Tsat, P and by the accuracy of the used equation of state. For the RKS equation of state the relative error eq 4 eq sat 3 ǫ(y T ) < 6 10− and ǫ(y P ) < 2 10− . The error in y is also induced c9 | sat × c9 | × c9 by the use of the RKS equation itself. To estimate this error, the phase-equilibrium RKS data should be compared with the experimental data at the conditions of in- eq terest. For typical saturator conditions, accurate data on yc9 is not available, so a direct estimation of the accuracy of RKS cannot be done. But for pressures higher than 150 bar there are some date available [19] and the relative error is within 3 5 10− . In Appendix C the saturation conditions and ﬂow values are reported ×

35 CHAPTER 2. WAVE TUBE EXPERIMENTAL METHOD for all experiments, so the mixture composition, in principle, can be recalculated with other equations of state.

2.6 Experimental procedure

A series of experimental runs is performed to obtain nucleation isotherms at given temperatures and pressures. A nucleation isotherm consists of 7 10 data points − taken at different supersaturations. From (A.15) it follows that the supersaturation can be changed by varying the vapor molar fraction yc9 or/and by varying eq eq the equilibrium molar fraction yc9 . The value of yc9 depends on the initial carrier eq gas mixture composition, and so variation in the composition will change yc9 . In a series, the nucleation temperature and pressure are not exactly the same, because the initial conditions slightly vary. Also, the non-ideal rupture of the diaphragm causes ﬂuctuations in the nucleation conditions. The reproducibility of the nucleation temperature is within 0.5 K and for the nucleation pressure it is within ± 0.2 bar. ±

Before a series is started, the pressure transducers and mass ﬂow controllers are calibrated. The details about the calibration procedures are in Sec. 2.3 and Sec. 2.5. The results of the calibrations are recorded and are compared with the earlier calibrations for the same transducers and controllers. The comparison allows to evaluate a dependence of calibration characteristics on time and to exclude the possibility of signiﬁcant errors during calibrations.

A particular experiment of a series begins with the evacuation of the high and low pressure sections. The sections are evacuated for 40-60 minutes, the final pres- 4 sure is less than 10− bar. After that, the mass flow control unit is set to obtain the desired mixture composition at the exit from the saturation section. Then, the saturation section is connected through the heated box to the high pressure section. The pressure of the gas-vapor mixture in the high pressure section is lower than in the saturation section. Therefore, to exclude the possibility of vapor condensation because of Joule-Thomson “cooling”, the pressure is reduced in the heat box at a temperature of 70 ◦C. The high pressure section is flushed with the gas-vapor mixture for 1 2 hours, to ensure that the process of vapor adsorption on the tube − wall has reached equilibrium, so that the composition of the flushing mixture is the same at the entrance to and exit from the high pressure section. In some experimental runs, it is examined whether the flushing time is enough to get adsorption equilibrium. The samples of the gas-vapor mixture are taken at the entrance to and at the exit from the high pressure section – the samples are analyzed with the gas-chromatograph. The equilibrium is reached when the GC-analysis of both samples gives the same results. The scheme of flushing is shown in Fig. 2.15. Af-

36 REFERENCES

Fig. 2.15: Scheme of the gas flow during flushing. The mass flow controller MFC adjusts the flow through the high pressure section HPS, so that the pressure is kept constant. By opening the appropriate valves, mixture samples can be taken at the entrance and at the exit from the high pressure section for gas-chromatographic GC analysis. By closing valves V1 and V2 the high pressure section is disconnected from the rest of the setup. ter flushing, the high pressure section is disconnected by closing inner valves V1 and V2 from the saturation section. The initial conditions (temperature, pressure, flows) are recorded, and than the diaphragm is ruptured. The diaphragm rupture is synchronized with the Le Croy wave-recorder which records and stores pressure and optical signals for further analysis. After the experiment the tube is opened, the remains of the diaphragm are removed, the new diaphragm is installed, and the next experiment can be started.

In the experiments the following values are directly measured: pressure, scattered and transmitted light intensities, the initial temperature of the high pressure section, the temperature and pressure in the saturators, and the ﬂows through the mixture preparation section. From these data the nucleation rate, nucleation temperature, pressure and supersaturation are derived.

References

[1] E.F. Allard and J.L. Kassner, Jr., J. Chem. Phys. 42, 1401 (1965).

[2] F.F. Abraham, Homogeneous Nucleation Theory, Academic Press, New York, 1974.

[3] Ya. B. Zel’dovich and Yu. P. Raizer, Physics of Shock Waves and High- Temperature Hydrodynamic Phenomena, Dover Publications, 2002.

37 CHAPTER 2. WAVE TUBE EXPERIMENTAL METHOD

[4] F. Peters, Exp. Fluids 1, 143 (1983).

[5] K.N.H. Looijmans, P.C. Kriesels, and M.E.H. van Dongen, Exp. Fluids 15, 61 (1993).

[6] K.N.H. Looijmans and M.E.H. van Dongen, Exp. Fluids 23, 54 (1997).

[7] K.N.H. Looijmans, Homogeneous nucleation and droplet growth in the coexistence region of n-alkane/methane mixtures at high pressures, PhD thesis, Eindhoven University of Technology, 1995.

[8] Chester L. Nachtigal (ed.), Instrumentation and Control: Fundamentals and Ap- plications, Wiley-IEEE, 1990.

[9] C.C.M. Luijten, Nucleation and Droplet Growth at High Pressure, PhD thesis, Eindhoven University of Technology, 1998.

[10] R.C. Reid, J.M. Prausnitz, and B.E. Poling, The Properties of Gases and Liquids, McGraw–Hill Book Company, New York, 1987.

[11] H.C. Van de Hulst, Light scattering by small particles, Dover, New York, 1981.

[12] M. Kerker, The scattering of light and other electromagnetic radiation, Academic Press, New York, 1969.

[13] C.F. Bohren and D.R. Huffman, Absorption and scattering of light by small particles, Wiley, New York, 1983.

[14] N.L. Swanson, B. D. Billard, and T. L. Gennaro, Applied Optics 38, 5887 (1999).

[15] W. B. Li, P. N. Segrè, R. W. Gammon, J. V. Sengers, and M. Lamvik, J. Chem. Phys. 101, 5058 (1994).

[16] R.C. Weast (ed.), Handbook of Chemistry and Physics, CRC Press, Cleveland, Ohio, 1976.

[17] P. Laven, MiePlot: A computer program for scattering of light from a sphere using Mie theory and the Debye series., www.philiplaven.com/mieplot.htm, 2006.

[18] J. Hrubý, New mixture-preparation device for investigation of nucleation and droplet growth in natural gas-like systems, Internal report, R-1489-D, Eindhoven Uni- versity of Technology, 1999.

[19] L.M. Shipman and J.P. Kohn, J. Chem. Eng. Data 11, 176 (1966).

38 Chapter 3

Nucleation and droplet growth

3.1 Kinetics of homogeneous nucleation

Unary kinetics

The process of cluster formation is a statistical process of detachment and attachment of clusters to each other. For example, in the unary two-phase system at thermodynamic equilibrium, clusters of different sizes will be formed in the gas phase. The probability to ﬁnd in that system a cluster consisting of n molecules will be proportional to the Boltzmann factor exp( W /k T ), where W is the for- − n b n mation energy of the given cluster. At constant conditions, the cluster distribution will be constant in time, but the clusters itself are in a dynamic equilibrium – they constantly break into parts and form again. Different situations can be imagined: a cluster can catch or lose a single vapor molecule, as well as two clusters can collide with each other forming a larger cluster, or the cluster can fall apart forming two smaller clusters.

The probability for a cluster to break apart or to collide with another cluster is much smaller than the probability to detach and catch a single molecule. The number density of clusters of any size is much smaller than the number density of a single vapor molecule, so a single molecule or a monomer cluster will collide with clusters more often than larger clusters do. If clusters are in a dynamic equilibrium, the detailed-balance law holds, so the rate of detachment of a single vapor molecule should be equal to the rate of attachment. It is assumed that the detachment or evaporation rate depends only on the cluster size, therefore the probability CHAPTER 3. NUCLEATION AND DROPLET GROWTH for a cluster to fall apart will be much lower than the probability to let evaporate a single molecule. Cluster-cluster interactions and cluster disintegration can be ex- cluded as unlikely events in comparison with cluster-monomer interactions, and the evaporation of a single molecule from a cluster.

In the supersaturated state, the kinetics of cluster interactions differs from that in thermodynamic equilibrium. First, the cluster size distribution changes with time. In the beginning, when the vapor is just brought into a supersaturated state, the cluster size distribution starts to evolve towards the equilibrium cluster size distribution. Condensation will appear which is maintained by a flow of clusters in cluster size space – the clusters are moving from smaller to larger ones. At some point, it can happen that the flow of the clusters will be the same for all cluster sizes, the steady-state nucleation case. The initial stage of condensation: the formation of a quasi-steady cluster distribution, usually takes 1 to 10 µs [1]. The characteristic time is determined by the impingement rate of vapor molecules with clusters. Steady state nucleation is possible because the amount of clusters with sizes larger than unity is much smaller than the number of single vapor molecules, so the vast amount of single molecules will provide a cluster flow in size space which is constant or almost constant for some period of time. When the source of single molecules is depleted, the supersaturation of vapor approaches unity, and the flow of clusters in size space will decrease. Ultimately, phase equilibrium will be achieved and the condensation process will stop; the cluster distribution will be equal to the equilibrium distribution.

Fig. 3.1: Cluster flow in size space. The condensation coefficient Cn and the evaporation En coefficient refer to for clusters of n-molecule size. The net rate Jn of evaporation and condensation defines the flow of clusters from size n to n + 1, see (3.2).

In Fig. 3.1 a kinetic scheme is shown for unary nucleation. The change of the molar cluster density ρn for n-molecule clusters is defined by the "inflow" in size space Jn 1 and the "outflow" Jn: −

dρn = Jn 1 Jn. (3.1) dt − −

The cluster ﬂow Jn is a net rate of condensation Cnρn of single molecules on clusters containing n molecules and of evaporation En+1ρn+1 of single molecules from

40 3.1. KINETICS OF HOMOGENEOUS NUCLEATION clusters containing n + 1 molecules:

J = C ρ E ρ . (3.2) n n n − n+1 n+1

The condensation coefﬁcient Cn can be deﬁned as a product of surface area sn, the impingement rate of monomers ζ, and the sticking probability for monomers P to attach to a cluster after impingement. From the gas kinetics laws, ζ can be expressed as

kbT ζ = ρ1 , (3.3) r2πM where ρ1 is the molar density of the monomers. The surface of the cluster can be 2/3 written as sn = an , where a is the cluster surface per molecule. For the sake of simplicity, we accept the usual assumption of perfect sticking: = 1. The P condensation coefﬁcient can be expressed as

2/3 kbT C = an ρ1 . (3.4) r2πM

It is rather difficult to evaluate the evaporation coefficient E directly. Usually, it is assumed that the evaporation rate is a function of cluster size and temperature, and it is not affected by the non-equilibrium character of the nucleation process. Therefore, at constant temperature, the evaporation coefficient at thermodynamic equilibrium will be equal to the evaporation rate at supersaturation conditions for clusters of the same size of n molecules:

eq En = En . (3.5)

At thermodynamic equilibrium the detailed-balanced conditions are fulﬁlled,

eq eq eq eq Cn ρn = En+1ρn+1, (3.6) which expresses that the net rate of cluster ﬂow in size space is equal to zero. For steady-state nucleation, the net ﬂow rate Jn in size space is independent of cluster size and subscript n can be left out. Combining equations (3.2), (3.5), and (3.6), the nucleation rate J can then be written as

J Cnρn ρn+1 eq eq = eq eq eq . (3.7) Cn ρn Cn ρn − ρn+1

The condensation coefﬁcient Cn is proportional to the monomer molar density (3.4). The monomer molar fraction is about the same as the vapor molar fraction, i.e. for diluted mixtures the following approximation holds:

Cn ≅ y eq eq = S, (3.8) Cn y

41 CHAPTER 3. NUCLEATION AND DROPLET GROWTH where the deﬁnition of supersaturation S from Sec. A.3 is used. Equation (3.7) is further modiﬁed by dividing both sides by Sn+1, which results in

J ρn ρn+1 eq n = eq n eq n+1 . (3.9) Cnρn S ρn S − ρn+1S

In the rhs of the expression, there are successive n and n + 1 terms. Therefore after summing the expressions written for n from 1 to N, the successive terms will be canceled out. The result of the summation is

N 1 ρN+1 J eq n = 1 eq N+1 . (3.10) C ρn S − ρ S n=1 n N+1 X For sufﬁciently large N the last term in the rhs is much smaller than unity and it can be omitted. Then the nucleation rate at steady-state conditions can be written as 1 ∞ 1 − J = eq n , (3.11) C ρn S n=1 n ! X where N in the summation limit is replaced by . If in the sum, the main contri- ∞ bution comes from terms with sufﬁciently large n, then it is possible to work with continuous variables instead of discrete ones. The steady state nucleation rate can be written as an integral:

1 − ∞ dn J = . (3.12)  C(n) ρeq(n) Sn  Z1  

The expressions for the nucleation rate (3.11) or (3.12) consist of quantities which can, in principle, be found. It is important to notice that the expression includes the eq density distribution ρn of clusters at thermodynamic equilibrium. So, to evaluate the nucleation rate, the supersaturation of the vapor, the condensation rate and the equilibrium cluster distribution should be known. The first two can be easily calculated in most cases, the last one is the most difficult part of the evaluation. Also, it should be mentioned that to calculate the nucleation rate with (3.11) or (3.12) there is no need to introduce the concept of a critical cluster. The main difficulty in the determination of the equilibrium cluster distribution lies in defining the energy of cluster formation for arbitrary size n. That question will be addressed in the following sections.

Nucleation kinetics in a carrier gas or carrier gas mixture

The kinetics of nucleation in a gas-vapor mixture is similar to that of supersaturated pure vapor. The only difference is that the cluster consists of two kinds

42 3.1. KINETICS OF HOMOGENEOUS NUCLEATION of molecules – vapor and gas molecules, and so the cluster now moves into two- dimensional size space. As for the unary case, it is supposed that the only important interactions are cluster collisions with gas and vapor monomers and the evaporation of a single gas or vapor molecule from the cluster. The cluster moves in size space by catching or losing single molecules. The kinetic scheme for this case is shown in Fig. 3.2, where nv and ng indicates the number of vapor and gas molecules in the cluster.

Fig. 3.2: Cluster ﬂows in two dimensional size space.

The temporal change of the number density of clusters with a given size (nv,ng) is expressed as

dρnv ,ng v v g g = Jnv 1,ng Jnv ,ng + Jnv ,ng 1 Jnv ,ng , (3.13) dt − − − − where the superscripts v and g indicates that a net cluster ﬂow is either in vapor or in gas direction in size space. In a similar way as for the unary net rate of condensation and evaporation, both the gas and vapor net rate can be written as

J v = Cv ρ Ev ρ (3.14) nv ,ng nv ,ng nv ,ng − nv +1,ng nv +1,ng and J g = Cg ρ Eg ρ . (3.15) nv ,ng nv ,ng nv ,ng − nv ,ng +1 nv ,ng +1 v The net rate Jnv ,ng of clusters in the v-direction is determined by evaporation and

43 CHAPTER 3. NUCLEATION AND DROPLET GROWTH condensation of vapor molecules from and to the cluster and it represents the net transition of clusters from (nv,ng)-mers to (nv + 1,ng)-mers. Similarly, the net rate g Jnv ,ng in the g-direction is determined by evaporation and condensation of gas molecules, and it deﬁnes the net transition from (nv,ng)-mers to (nv,ng + 1)-mers. The condensation coefﬁcients for gas and vapor molecules are

k T Cv = s ρ b , (3.16) nv ,ng nv ,ng 1,0 2πM r v and

g kbT Cnv ,ng = snv ,ng ρ0,1 . (3.17) s2πMg Supposing that the cluster consists of a homogeneous liquid mixture and that the cluster is spherical, its surface area snv ,ng can be determined as

1/3 liq liq 2/3 snv ,ng =(36π) nvv˜v + ngv˜g , (3.18) liq liq where v˜v and v˜g are the partial molar volumes of vapor and gas components in the liquid phase.

To compare the condensation coefﬁcients of vapor and gas molecules, expression (3.17) is divided by (3.16) which gives

g Cnv ,ng ρ0,1 Mv v = . (3.19) Cnv ,ng ρ1,0 sMg

The molecular weight Mv of the vapor component is usually larger than the molecular weight Mg of the gas component. For example for hydrocarbons, the condensation temperature is decreasing with molar mass of a hydrocarbon. The molecular number densities of monomers ρ1,0 and ρ0,1 are approximately equal to the molecular number densities of the gaseous components in the mixture. So, for the mix- g ture with low vapor molar fraction the condensation coefficient Cnv ,ng significantly v g exceeds Cnv ,ng . The flow rate Jnv ,ng in ng direction in size space will be of the same order or less than the flow rate J v in n direction: J g = J g . nv ,ng g O nv ,ng O nv ,ng v The cluster flow in nv direction is of the order of Cnv ,ng ρnv ,ng , and because the gas condensation coefficient significantly exceeds the vapor condensation coefficient it follows that Cg ρ Eg ρ . (3.20) nv ,ng nv ,ng ≈ nv ,ng +1 nv ,ng +1

In full phase equilibrium, there is a dynamic balance between condensation and evaporation, and the following equality holds:

g,eq eq g,eq eq Cnv ,ng ρnv ,ng = Env ,ng +1ρnv ,ng +1. (3.21)

44 3.1. KINETICS OF HOMOGENEOUS NUCLEATION

Because of the high molar fraction of the gas component, the change of the vapor molar fraction from supersaturated to equilibrium state almost does not change the impingement rate of the gas molecules with a cluster (n ,n ), so Cg,eq Cg . v g nv ,ng ≈ nv ,ng Also, as in the unary case, it is assumed that the evaporation rate is determined g g,eq only by the cluster size Env ,ng = Env ,ng . Then (3.20) can be divided by (3.21), which gives ρnv ,ng ρnv ,ng +1 eq eq . (3.22) ρnv ,ng ≈ ρnv ,ng +1 The ratio of the equilibrium and supersaturated cluster distributions does not depend on the number of gas molecules in the cluster.

Fig. 3.3: Directions of cluster ﬂows in two dimensional ng-nv size space.

g The cluster ﬂow Jnv ,ng in ng-direction should decrease with an increase of the amount of ng molecules in the cluster. This is because the gas component is not condensable, therefore clusters with large amount of gas molecules cannot be formed.

The vector of cluster ﬂow Jnv ,ng in size space should eventually turn into the nv direction, as it is indicated in Fig 3.3. So, for quasi-steady nucleation, the nucleation v rate can be deﬁned as the sum of all nucleation rates Jnv ,ng for clusters containing nv molecules: v ∞ v J = Jtot = Jnv ,ng . (3.23) n =0 Xg

Now, using the detailed balance conditions for cluster movement in nv direction

45 CHAPTER 3. NUCLEATION AND DROPLET GROWTH at thermodynamic equilibrium

g,eq eq g,eq eq Cnv ,ng ρnv ,ng = Env +1,ng ρnv +1,ng (3.24) and the fact that the evaporation rate is the same in equilibrium and in the su- v v,eq persaturated state, Env ,ng = Env ,ng , the following expression can be derived from (3.14): v v Jnv ,ng Cnv ,ng ρnv ,ng ρnv +1,ng v,eq eq = v,eq eq eq . (3.25) Cnv ,ng ρnv ,ng Cnv ,ng ρnv ,ng − ρnv +1,ng To calculate the nucleation rate, (3.25) is rearranged and combined with (3.23). Then after summation from from n = 0 to , it follows that g ∞

v ∞ C ρ v,eq eq nv ,ng nv ,ng ρnv +1,ng J = Cnv ,ng ρnv ,ng v,eq eq eq . (3.26) Cn ,n ρn ,n − ρ n =0 v g v g nv +1,ng ! Xg For a small vapor concentration, the vapor supersaturation can be deﬁned as a ratio of vapor molar fraction in the supersaturated and in the equilibrium state (A.15). Taking into account (3.16) and the fact that the vapor molar fraction is almost equal to the vapor monomer molar fraction, the vapor supersaturation can be calculated v v,eq as Sv = Cnv ,ng /Cnv ,ng . Because the term between parentheses in (3.26) is independent of ng (3.22), the nucleation rate expression (3.26) can be rewritten to the form J ρnv ,ng Sv ρnv +1,ng v,eq eq = eq eq . (3.27) ∞ C ρ ρ ρ ng =0 nv ,ng nv ,ng nv ,ng − nv +1,ng

Further, both partsP of (3.27) are divided by Snv +1

J ρn ,n ρn +1,n = v g v g . (3.28) v eq nv eq nv eq nv +1 ∞ C ρn ,n Sv ρn ,n Sv − ng =0 nv ,ng v g v g ρnv +1,ng Sv P Summing from nv = 0 to Nv gives

Nv 1 ρN +1,n J = 1 v g (3.29) v eq nv eq Nv +1 ∞ C ρnv ,ng Sv − ρ S n =1 ng =0 nv ,ng ! Nv +1,ng v Xv P where due to mutual cancellations in the right side of the equation only the ﬁrst and the last term of the sum are present. For a sufﬁciently large Nv, the last term

eq Nv +1 ρNv +1,ng / ρNv +1,ng Sv is very small, and this term can be left out of the equation. After that, the expression for the nucleation rate is easy to obtain, and it is 1 ∞ 1 − J = . nv v eq (3.30) Sv ∞ C ρn ,n n =1 ng =0 nv ,ng v g !! Xv P

46 3.1. KINETICS OF HOMOGENEOUS NUCLEATION

In some cases the sums in (3.30) can be replaced by integrals

1 − ∞ 1 J =   dnv . (3.31) nv ∞ v eq Z Sv C (n , n ) ρ (n , n ) dn 1  v g v g g     0     R  

The expression (3.30) for the nucleation rate in a gas-vapor mixture resembles the expression for unary nucleation (3.12). By introducing an effective vapor con- eff eff densation rate Cnv and an effective vapor molar number density ρnv , which are deﬁned as v eq ∞ C ρ eff ng =0 nv ,ng nv ,ng Cnv = eq (3.32) ∞ ρn ,n P ng =0 v g and P ∞ eff eq ρnv = ρnv ,ng , (3.33) n =0 Xg the expression for the nucleation rate in gas-vapor mixtures can be reduced to a quasi-unary form: 1 ∞ 1 − J = . eff eff nv (3.34) C ρ Sv n =1 nv nv ! Xv In this study, the obtained quasi-unary expression (3.34) for the nucleation rate is the basic expression for further analysis of nucleation in gas-vapor mixtures.

It is quite straightforward to extend nucleation rate expression (3.30) to mixtures containing one vapor component and several non-condensable gases. For example, for ternary mixture consisting of one vapor component and two non- condensable gas components, the nucleation rate expression is

1 ∞ 1 − J = , nv v eq (3.35) Sv ∞ ∞ C ρn ,n 1,n 2 n =1 ng 1=0 ng 2=0 nv ,ng 1,ng 2 v g g !! Xv P P where ng1 and ng2 are the numbers of first and second gas molecules in a cluster. In deriving nucleation rate expression (3.35), it is essential that the vapor molar fraction is significantly less than the molar fractions of first and second gas components.

47 CHAPTER 3. NUCLEATION AND DROPLET GROWTH

3.2 Nucleation theories

Classical nucleation theory

According to (3.11), in order to determine the quasi-steady nucleation rate, the cluster size distribution at equilibrium conditions should be known. For a unary system, the cluster distribution is proportional to the Boltzmann factor [1]:

W ρ = ρ exp n , (3.36) n 1 −k T b where the proportionality coefﬁcient is equal to the monomer number density. Be- cause at equilibrium the chemical potentials of gas and liquid phases are the same, the bulk term n µliq µgas disappears in the expression for the free energy of − cluster formation and only the surface term will stay. Then the free energy of cluster formation in phase equilibrium is given by the energy of surface formation 2/3 Wn = sσnn , (3.37)

1 where s is the surface of the cluster per molecule and σn is the "surface tension" for the n molecule cluster. The "surface tension" of an n-cluster will, in general, be dependent on cluster size. In the limit of very large clusters, n , its value → ∞ becomes the ﬂat interface surface tension σ . Knowledge on the size-dependence ∞ of surface tension is limited. For large clusters, this dependence is described by the Tolman correction [2, 3]. Recently Kalikmanov [4] has proposed a new semi- phenomenological theory for the surface energy of clusters. Here, it is assumed that the surface tension is the same for clusters of all sizes and it is equal to the surface tension σ of the gas-liquid interface at equilibrium. ∞

Now, the nucleation rate expression (3.11) can be rewritten as

1 ∞ 1 − J = , (3.38) C ρeq exp θ n2/3 + n ln S n=1 n 1 ! X − ∞ where θ is the dimensionless surface tension: ∞ sσ θ = ∞ . (3.39) ∞ kbT

In the nucleation rate expression (3.38) all variables are known and the nucleation rate J can be found just by calculating the sum. But it is possible to simplify the expression, because the main contribution in the sum comes from the largest term and terms nearby which are determined by the value of the exponent, the depen-

1The surface tension is deﬁned for the phase interface at equilibrium, so for clusters it is more accurate to speak about the energy of surface formation.

48 3.2. NUCLEATION THEORIES

dence of the condensation coefﬁcient Cn on the cluster size being relatively weak. In order to evaluate the terms which contribute most to the nucleation rate, the sum in (3.38) is ﬁrst transformed into an integral:

1 − ∞ dn J = . 2/3 (3.40)  C(n) ρ1 exp θ n + n ln S  Z1 − ∞   In the argument of the exponent there are two terms; for small values of n, the ﬁrst, surface term will prevail, and for large values of n the second term will win. Because the terms have different signs, the argument of the exponent has a minimum at some value n∗. It appears that the argument equals the energy of cluster formation, W n = n ln S + θ n2/3, (3.41) kbT − ∞ in the supersaturated state at quasi-steady nucleation. It means that n∗ is indeed equal to the number of molecules in the critical cluster as it is introduced in the classical nucleation theory [5]. Then the size of the critical cluster can easily be found from (3.41) with the formation energy of the critical cluster:

3 W ∗ 4 θ = ∞ 2 . (3.42) kbT 27(ln S)

The same expression for the energy of critical cluster formation was obtained in Sec. 1.1 from the thermodynamic point of view on cluster formation.

The free energy Wn in the integral (3.45) can be replaced with a second order

Taylor expansion nearby of n∗:

2 2 W W ∗ Z πk T (n n∗) , (3.43) n ≈ − b − with Zeldovich factor Z given by

2 1/2 1 ∂ Wn 1 θ 2/3 ∞ − Z = 2 = (n∗) . (3.44) −2πk T ∂n ∗ 3 π b n=n

The condensation coefﬁcient C(n) in (3.45) varies slowly with n in comparison eq n with the product ρ(n) S , therefore it can be taken at n = n∗and it can be put in front of the integral:

1 ∞ − eq W ∗ 2 2 J = C(n∗) ρ1 exp exp Z πkbT (n n∗) dn . (3.45) −kbT  − −  Z1   The integral in (3.40) has a Gaussian form. By extending its interval of integration to ( , ), its numerical value can be easily evaluated. Taking into account the −∞ ∞ deﬁnition of the condensation coefﬁcient (3.4) together with (3.42) and (3.44), the

49 CHAPTER 3. NUCLEATION AND DROPLET GROWTH

ﬁnal expression for the unary nucleation rate can then be written in the following form 1/2 3 eq liq 2σ 4 θ J = ρ ρ v ∞ exp ∞ . (3.46) 1 1 v πM −27 2 (ln S) ! The nucleation rate expression (3.46) is based on the classical nucleation theory and on the capillarity approximation, replacing the cluster properties like density and surface tension with the properties of the corresponding liquid phase at equilibrium.

Quasi-unary nucleation theory

Now let us consider quasi-steady nucleation in a supersaturated gas-vapor mixture. Again, as for the unary case, to evaluate the quasi-steady nucleation rate eq the equilibrium cluster distribution ρnv ,ng should be known. Then it should be inserted in (3.30), and the value of nucleation rate J will be obtained after performing summation.

The equilibrium cluster distribution for the binary gas-vapor mixture can be written as s xeq xeq Wn ,n ρeq = ρeq v ρeq g exp v g , (3.47) nv ,ng 1,0 0,1 − k T b where ρ1,0 and ρ0,1 are number densities of vapor and gas monomers at equilib- eq eq rium, xv and xg are equilibrium molar fractions of vapor and gas components in s the bulk liquid phase, and Wnv ,ng is the surface energy of an (nv,ng)-cluster formation. Expression (3.47) can be derived from the general expression for the cluster distribution in a binary mixture suggested by Wilemski and Wyslouzil [6].

Further as in the unary case, the capillarity approximation is used, namely it is assumed that the surface tension of an arbitrary cluster equals the surface tension σ ∞ for a ﬂat gas-liquid interface at equilibrium. Then the energy of surface formation can be rewritten to the form

s Wn ,n = snv ,ng σ . (3.48) v g ∞

The surface of the cluster can be evaluated with (3.18).

For the sake of simplicity, the approximated integral expression (3.31) will be used to evaluate the nucleation rate J instead of the more precise expression (3.30). For the known equilibrium cluster distribution, the inner integral can be calcu-

50 3.2. NUCLEATION THEORIES lated

∞ v eq C (nv, ng) ρ (nv, ng) dng = Z0 ∞ 1/3 liq liq 2/3 2/3 (36π) nvv˜v + ngv˜g σ A n v˜liq + n v˜liq exp ∞ dn , (3.49) v v g g − k T g Z b ! 0 where a parameter A is introduced, that does not depend on the number ng of gas molecules in the cluster

xeq xeq k T A =(36π)1/3 ρ ρeq v ρeq g b . (3.50) 1,0 1,0 0,1 2πM r v The integral expression in the rhs of Eq. (3.49) can be simpliﬁed by introducing the following notation: liq liq z = nvv˜v + ngv˜g , (3.51)

(36π)1/3 σ k = ∞ , (3.52) kbT and liq b = nvv˜v . (3.53)

The value of the simpliﬁed integral can be found in mathematical handbooks, and it is

∞ z2/3 exp kz2/3 dz = − Zb 9 b1/3 + 6 b k 9 √π erfc(b1/3 √k) exp b2/3 k + . (3.54) 4 k2 − 8 k5/2 At typical conditions, the first term in the rhs of (3.54) is a few orders of magnitude larger than the second term which includes the complementary error function erfc. Furthermore, the ratio between the two terms only increases as the cluster consists of more and more vapor molecules. For example, for a methane/n-nonane mixture at 240 K and 40 bar, the first term is more than 200 times larger than the second one. The prefactor of the first term has two parts: 9b1/3 and 6bk. The part proportional to b1/3 exceeds bk only if the cluster is very small; this condition is even stronger because k is a very large number as follows from its definition (3.52). As a result, the integral can be approximated with high accuracy by the form

∞ 3 b z2/3 exp kz2/3 dz exp b2/3 k . (3.55) − ≈ 2 k − Zb

After the inner integral in (3.31) being taken, the expression for the nucleation

51 CHAPTER 3. NUCLEATION AND DROPLET GROWTH rate can be rewritten as:

1 − ∞ dn J = v . (3.56) 3 b A 2/3  2 k liq exp b k + nv ln Sv  Z1 v˜g −   By analogy with the unary nucleation rate (3.40), the argument of the exponent can be interpreted as some effective energy of cluster formation or as a quasi-unary energy of cluster formation:

eff Wn 2/3 = nv ln Sv + θ nv , (3.57) kbT − ∞ where the dimensionless surface tension θ is deﬁned as ∞

1/3 liq 2/3 (36) v˜v σ θ = ∞ . (3.58) ∞ kbT

In the quasi-unary nucleation theory, the effective energy of nv-cluster formation has a similar form as the energy of n-cluster formation (3.41) in unary nucleation theory, but the dimensionless surface tension θ should be calculated using the ∞ phase equilibrium surface tension σ for the gas-vapor mixture and the partial ∞ liq molar volume v˜v of the vapor component in the liquid phase.

Following a similar procedure as was done for the derivation of the unary nucleation rate expression, taking into account only contributions of clusters which are close to the critical cluster, the expression for the nucleation rate in a binary gas-vapor mixture becomes:

eq eq 1/2 liq liq 3 eq xv eq xg 2 σ v˜v v˜v 4 θ J = ρ ρ ρ ∞ exp ∞ . (3.59) 1,0 1,0 0,1 πM liq ln S −27 2 v v˜g ! v (ln Sv) ! The quasi-unary nucleation rate expression is very similar to the unary nucleation rate expression (3.46). Furthermore, it is possible to extend (3.59) to mixtures which contain one vapor component and several non-condensable gas components. The derivation is similar to that used for a binary gas-vapor mixture. Here, the quasi- unary nucleation rate expression is speciﬁed for ternary gas-vapor mixtures:

eq eq eq 1/2 eq x eq x eq x 2 σ J = ρ ρ v ρ g1 ρ g2 ∞ 1,0,0 1,0,0 0,1,0 0,0,1 πM × v liq liq liq 3 v˜v v˜v v˜v 4 θ exp ∞ , (3.60) × liq liq ln S2 −27 2 v˜g1 ! v˜g2 ! v (ln Sv) ! based on (3.35).

The solubility of a non-condensable gas component in the liquid phase characterizes how strong such a component can change the energy of cluster formation.

52 3.2. NUCLEATION THEORIES

eq Therefore, if xg = 0, the non-condensable gas component does not inﬂuence the nucleation process, playing the role of an inert carrier gas which only "absorbs" the latent heat of condensation. It means that any binary nucleation theory should reduce to the unary theory in the limit xeq 0. The limit can be realized by low- g → ering the nucleation pressure or by replacing the gas component with one which poorly dissolves in the liquid phase.

The nucleation expressions (3.59) and (3.60) cannot be reduced to the unary nucleation rate expression (3.46) by taking the limit xeq 0. The problem originates g → from the formation energy of a cluster (3.48). The formation energy only includes the surface term, which is the same for well and poorly dissolving gases. The problem is solved by including the mixing entropy term in the energy of cluster formation:

nv ng Wnv ,ng = snv ,ng σ + nv ln eq + ng ln eq . (3.61) ∞ (n + n ) x (n + n ) x v g v v g g According to this expression, the energy of cluster formation with a composition different from the equilibrium composition is relatively high (see Fig. 3.4). Small clusters with a composition close to equilibrium are energetically favorable to be formed. In the limit xeq 0, the cluster which does not contain gas molecules has g → the lowest energy of formation. As a result, in the inner sum of (3.30) only the ﬁrst term with ng = 0 survives and the binary nucleation rate expression reduces to the unary expression (3.11).

160 cluster energy surface term 140 mixing term

120

100 T) b /(k

g 80 ,n v n

W 60

0 0 5 10 15 20 25 30 35 40 45 50

Fig. 3.4: Contribution of surface and mixing terms to the energy of formation for a eq methane/n-nonane cluster with nv = 20 at 240 K and 40 bar,(xg = 0.26).

The quasi-unary nucleation rate expression (3.30) with an appropriate expression for the energy of cluster formation provides a general approach to the description of nucleation in gas-vapor mixtures.

53 CHAPTER 3. NUCLEATION AND DROPLET GROWTH

3.3 Nucleation theorem

The concept of a critical cluster is introduced and utilized in most nucleation theories. The critical cluster is in metastable equilibrium with the vapor; for a cluster with a size larger than critical, the liquid phase is thermodynamically favorable, so the formation of critical clusters determines the nucleation rate. To verify nucleation theories and to develop them further, the size of the critical cluster should be known. The critical cluster is a microscopic object expected to be tens of molecules in size for the conditions studied. Because critical clusters are so small, it is difﬁcult to measure their sizes. Only if complicated and expensive measurement techniques, like neutron scattering [7], are utilized – direct measurements can be performed. In most experiments, macroscopic droplets are detected, and from these measurements the nucleation rate data is inferred. The nucleation theorem provides a convenient way to estimate the size and the composition of the critical cluster on the basis of nucleation rate data in a theory-independent fashion.

In Sec. 1.1 it was shown that for the classical nucleation theory, the formation gas energy W ∗ of a critical cluster and the difference of chemical potentials ∆µ between an initial (supersaturated) and phase equilibrium in the gas phase are related to the size of the critical cluster n∗

dW ∗ = n∗. (3.62) d∆µgas −

Kashchiev [8] proved (3.62) to be of general validity if n∗ is substituted with the excess number of molecules ∆n∗ in the critical cluster. The relation in this form can be derived without reference to any particular nucleation theory, and does not depend on the size of a critical cluster. Furthermore, the expression is valid for a wide range of nucleation phenomena (gas-to-liquid, liquid-to-liquid, liquid-to- solid). But in nucleation experiments the nucleation rate is measured, not the energy of cluster formation. The nucleation rate J in a vapor can be written as

W ∗ J = A exp , (3.63) −k T b and the generalized supersaturation S can be deﬁned as

∆µgas S = exp . (3.64) k T b The pre-exponential kinetic factor A is a weak function of supersaturation, so partial derivation of (3.63) with respect to supersaturation at constant temperature and pressure results in ∂ ln J = ∆n∗. (3.65) ∂ ln S − T,P Equation 3.65 is suitable for an analysis of nucleation rate data: if the data are in the

54 3.3. NUCLEATION THEOREM form of nucleation isotherms, the slope of an isotherm yields the number of molecules in the critical cluster. Sometimes (3.65) is also referred to as the nucleation theorem.

The theorem (3.62) for unary vapor nucleation has been generalized and further extended by Oxtoby and Kashchiev [9] to multicomponent systems. They start from the basic expression for the energy of cluster formation and arrived to the following relation ∂W ∗ gas = ∆ni∗, (3.66) ∂µ0,i − gas which links the energy of critical cluster formation, the chemical potential µ0,i of the i-th component in the initial gas-vapor mixture and the excess number ∆ni∗ of i-th molecules in the critical cluster. The expression can be applied not only to vapor-to-liquid nucleation but to a wider range of nucleation phenomena, and it holds for clusters of arbitrary size.

With the extended nucleation theorem, the nucleation rate data in the binary mixture can be analyzed. Nucleation in the presence of a carrier gas can be treated as a binary nucleation process. The nucleation theorem is then applied to study the inﬂuence of a carrier gas on nucleation. In a gas-vapor mixture, the nucleation rate is determined by the supersaturation of the vapor component Sv or by the vapor molar fraction y0,v, pressure P and temperature T . From the general expression for the nucleation rate (3.63), the set of equations after applying the nucleation theorem becomes:

∂ ln J = ∂ ln S v P,T gas gas (3.67) ∆n ∂µ ∆n∗ ∂µ ∂ ln A v∗ 0,v + g 0,g + k T ∂ ln S k T ∂ ln S ∂ ln S b v P,T b v P,T v P,T and

∂ ln J = ∂ ln P Sv ,T gas gas (3.68) ∆n ∂µ ∆n∗ ∂µ ∂ ln A v∗ 0,v + g 0,g + . kbT ∂ ln P kbT ∂ ln P ∂ ln P Sv ,T Sv ,T Sv ,T The partial derivatives in the lhs can be found from the nucleation rate data if nucleation isotherms are available in a sufﬁciently wide range of supersaturation and pressure. The partial derivatives in the rhs can be estimated from the equilibrium thermodynamic properties of the liquid and gas bulk phases. Then the numbers of excess vapor and carrier gas molecules can be obtained by solving (3.67-3.68). Such analysis of nucleation rate data obtained in a carrier gas has been performed by Oxtoby and Laaksonen [10], Luijten et al. [11] and others. Here an approach suggested by Luijten is adopted to simplify the equations.

55 CHAPTER 3. NUCLEATION AND DROPLET GROWTH

On the basis of a kinetic model of nucleation, the pre-exponential kinetic factor A can be found. It is generally accepted that the kinetic factor is proportional to the gas gas product of molar densities in the initial state and in the equilibrium state: ρ0 ρeq , recalling that S ρgas/ρgas, and that the equilibrium density can be expressed as v ≈ 0 eq sat gas fePv / (Z kbT ) if the vapor molar fraction in gas-vapor mixture is small. The pre-exponential factor also is a function of surface tension, but the surface tension depends on pressure to a less extent. Therefore, at constant temperature the kinetic factor is expressed as f 2 A S e . (3.69) ∼ v Zgas gas The compressibility factor Z and the enhancement factor fe are functions of pressure and temperature only, so

∂ ln A = 1. (3.70) ∂ ln S v P,T

gas The chemical potential of the gas component µ0,g does not depend on supersaturation, when the fraction of the vapor component in the mixture is small:

∂µgas 0,g 0. (3.71) ∂S ≈ v P,T From the deﬁnition of supersaturation (3.64), it immediately follows that

∂µgas 0,v = k T, (3.72) ∂ ln S b v P,T

gas where it is taken into account that for binary mixtures, µeq,v is a function of temperature and pressure only. Now, substituting the partial derivatives (3.70-3.72) into (3.67), an expression relating the slope of the nucleation isotherms with the excess number of vapor molecules can be written as

∂ ln J = ∆n∗ + 1. (3.73) ∂ ln S v v P,T

For the pressure part of the nucleation theorem (3.68), it is convenient to switch from the set of variables (Sv,P ,T )totheset(y0,v,P ,T ). Because of the exact thermo- (∂µ /∂P ) =v ˜ v˜ dynamic relation i yi,T i, i being the partial molar volume, the pressure part of the nucleation theorem can be rewritten as

gas gas ∂ ln J P v˜v P v˜g = ∆nv∗ + ∆ng∗ + 2 ln fe, (3.74) ∂ ln P kbT kbT y0,v ,T where it is used that in ﬁrst order approximation the natural logarithm of the en-

56 3.3. NUCLEATION THEOREM hancement factor is proportional to pressure:

∂ ln A ∂ ln Zgas = 2 ln f 2 2 ln f , (3.75) ∂ ln P e − ∂ ln P ≈ e y0,v ,T T where the second term in the rhs of the expression was skipped, because it is much smaller than the ﬁrst one.

The nucleation rates are determined experimentally in a rather narrow range and the dependence of nucleation rate on vapor molar fraction is very steep, so (∂ ln J/∂ ln P ) y0,v ,T can only be inferred from the data with large errors. But, the pressure part of the nucleation theorem can be modiﬁed, taking into account that from a mathematical point of view, the nucleation rate is a function of the independent variables y0,v, P , and T . Therefore their partial derivatives are related

∂ ln J ∂ ln y ∂ ln J = 0,v . (3.76) ∂ ln P − ∂ ln P ∂ ln y0,v y0,v ,T J,T P,T

At constant temperature and pressure, vapor supersaturation is proportional to the vapor molar fraction, so(∂ ln J/∂ ln y0,v)P,T can be replaced with(∂ ln J/∂ ln Sv)P,T , and (3.76) can be written as

gas gas P v˜v P v˜g ∂ ln y ∆nv∗ + ∆ng∗ + 2 ln fe 0,v = kbT kbT , (3.77) ∂ ln P − ∆n + 1 J,T v∗ where relation (3.73) is taking into account. How strong the gas component molecules which are present in the cluster will inﬂuence the nucleation rate depends gas on the value of the factor P v˜g /(kbT ). For a small molar concentration of the vapor component, the factor approximately equals the compressibility of the gas- vapor mixture Zgas in the gas phase. The compressibility factor is always positive, it characterizes the non-ideality of the gas-vapor mixture and the strength of the intermolecular interactions. At low pressures, the factor is close to unity , and it can decrease or increase when the pressure goes up [12]. So, the presence of gas component molecules in the cluster facilitates nucleation for higher pressures, and the effect depends on the compressibility factor. How the presence of vapor com- gas ponent molecules will inﬂuence nucleation is determined by the sign of v˜v . The vapor partial molar volume in the gas phase can be positive as well as negative, it depends on the interaction between the vapor and gas molecules. The enhancement factor fe is a measure of the deviation from Raoult’s law for the composition of the vapor component in the binary system at phase equilibrium. For low pressures it is close to unity, going up with pressure increase, and virtually stimulating nucleation.

The supersaturation part (3.73) and the pressure part (3.77) of the nucleation theorem consist of(∂ ln y0,v/∂ ln P )J,T and(∂ ln J/∂ ln Sv)P,T which are found from the

57 CHAPTER 3. NUCLEATION AND DROPLET GROWTH

gas gas experiments; fe, v˜g , and v˜v can be calculated with an appropriate equation of state. Then, the excess number of molecules ∆nv∗ and ∆nv∗ are determined as so- lutions of equations (3.73) and (3.77). The size and the composition of the critical cluster of a gas-vapor mixture is estimated with the extended nucleation theorem from the nucleation rate data and from the equilibrium bulk properties of the two- phase system.

3.4 Droplet growth

Condensation can be thought as being a two-stage process: nucleation and droplet growth. During nucleation, clusters spontaneously gain and lose molecules, with the net rate of losing and gaining depending on the energy of cluster formation. For the critical cluster both rates are the same; for sub-critical clusters, the losing rate prevails over the gaining rate, and the reverse holds for super-critical clusters. When a super-critical cluster reaches a certain size, the gaining rate overcomes signiﬁcantly the losing rate – spontaneous evaporation of the cluster is almost impossible. Muitjens [13] shows that this happens when the number of molecules in the cluster is twice of that of the critical cluster. Such super-critical clusters are called droplets, and the process of gaining molecules by these clusters is referred to as droplet growth.

Droplet growth is determined by the mass-transfer towards the droplet. But, in order to describe droplet growth correctly, it is not enough to know the mass- transfer, because the condensation is accompanied with heat release. Therefore, heat-transfer from the droplet should be considered as well. A full description of droplet growth includes both mass and heat ﬂows.

Mass and heat transfer for growing droplets can be described from two points of view: either using kinetic or fluid dynamics laws. Which point of view is most suitable depends on the value of the Knudsen number. The Knudsen number Kn 2 is the ratio of the mean free path lv of vapor molecules to the diameter of the droplet l Kn = v . (3.78) 2rd For large Knudsen numbers Kn 1, droplet growth is governed by molecular ki- ≫ netic laws: the mass and heat flows are characterized by the impingement rate of vapor molecules. For small Knudsen numbers Kn 1, droplet growth is gov- ≪ erned by the diffusion of vapor molecules toward the droplet and it can be described in the frame of continuum fluid dynamics. The Knudsen number is inversely proportional to the size of a droplet, so for a small droplet the kinetic description should be applied, but as the droplet is growing, it reaches a size for

2which mean free path has been addressed by Peeters et al. [14]

58 3.4. DROPLET GROWTH which the Knudsen number is small and the continuum description should be used. Apparently, an intermediate droplet growth regime occurs at Kn = (1). It O can be described with a theory which incorporates methods of molecular kinetics as well as continuum heat and mass transfer. The theories of kinetic and diffusion- controlled droplet growth are very well developed [15, 16, 17]. Droplet growth models for arbitrary Knudsen numbers are given, among others, by Gyarmathy [18] and Young [19].

In this study, droplet growth is observed for droplets 0.1 -1.0 µm in size, at pressures from 10 to 45 bar. The mean free path lv is inversely proportional to pressure, so according to (3.78), Kn is decreasing with pressure increase. Therefore, at higher pressures, diffusion-controlled droplet growth is expected for smaller droplets. At the experimental conditions, droplet growth is diffusion controlled and it is described with the continuum equations of mass and heat transfer. Furthermore, based on a series of assumptions, it is possible to derive a simple expression for the growth rate of a droplet.

Droplet growth model

The growing droplet is spherical and it is surrounded with a supersaturated gas- vapor mixture. As shown in Fig. 3.5, three spatial zones are distinguished for a growing droplet: (1) the interior of the droplet (all the variables referring to the interior have subscript d, like for the droplet temperature Td), (2) the region at the droplet surface (subscript s), and (3) the zone far from the droplet (subscript far). The far zone is at the distance rfar from the droplet, which can be deﬁned as: 3 1/3 r = , (3.79) far 4πn where n is the droplet number density. In the experiments performed, only in a very extreme case – high droplet number density and large droplet size – the ratio between rfar and rd is about 10; at typical conditions the ratio is much larger.

The droplets are far from each other, so they are growing independently without any interactions: no coalescence occurs. There is no slip between the droplet and the gas vapor mixture, they are not moving relative to each other: the gas-vapor mixture does not ﬂow around the droplet.

The droplet characteristics (pressure, temperature and composition) are assumed uniform in the droplet. Furthermore, the droplet is in mechanic and thermodynamic equilibrium with the gas-vapor mixture at the droplet surface. The surface of the droplet is moving slowly in comparison with the speed of sound, so any pressure ﬂuctuations are equalized very fast, and pressure is virtually constant

59 CHAPTER 3. NUCLEATION AND DROPLET GROWTH

Fig. 3.5: Droplet growth model. everywhere from the droplet surface to the far zone distance. In the experiments, the pressure itself deviates slowly with a characteristic time of the order of 10 ms, but the deviation is not sufficiently large to influence droplet growth appreciably. The vapor molar fraction in the gas-vapor mixture is small, of the order of hundreds ppm, but virtually the flow of vapor molecules determines the droplet growth rate.

Mass-transfer equation

The vapor and gas molecules are moving towards a droplet with an average molar velocity us in the laboratory frame ﬁxed to the droplet with the origin in the center of the droplet. When molecules reach the droplet surface – they get absorbed. As a result the size of the droplet is increasing, and its surface is moving with the velocity drd/dt. The molar ﬂow in radial direction at the droplet surface is

dr Φ = 4πr2ρgas d u , (3.80) − d s dt − s gas where ρs is the molar density of the gas-vapor mixture at the droplet surface. It is assumed that the droplet is growing at quasi-steady conditions, so the relax- ation time for density changes caused by the droplet growth is much less than the characteristic time of the droplet growth itself. As a result the molar ﬂow will be independent of radius: 2 gas 2 gas rdρs us = r ρ u, (3.81) where the gas-vapor density ρgas and molecular velocity u are taken at some distance r from the droplet center. The droplet molar mass is changing as

d 4 M˙ = πr3ρliq , (3.82) d dt 3 d d 60 3.4. DROPLET GROWTH

liq where ρd is the liquid molar density of the droplet. The molar ﬂow towards the droplet equals the rate of change of the droplet molar mass M˙ = Φ, so after com- d − bining (3.80) with (3.82) and (3.81) and some rearrangements, the equation

dr r2ρgasu = ρliq ρgas r2 d (3.83) − d − s d dt is obtained. In the derivation of this equation, it has implicitly been assumed that the liquid droplet is incompressible with a ﬁxed composition: the change of the liq droplet pressure caused by the change of the droplet size does not affect ρd . Fur- thermore, the rhs of the equation depends only on time. To make the notation sim- pler, a quantity F characterizing the total ﬂow toward droplet is introduced:

dr F ρliq ρgas r2 d . (3.84) ≡ d − s d dt

The molar density of the gas-vapor mixture can be written as

gas gas gas ρ = ρg + ρv , (3.85)

gas where ρg is the molar density of the gas component (it can be a mixture of non- gas condensable gases) and ρv is the molar density of the vapor component. The gas and vapor molecules are moving in the gas-mixture relatively to each other; their diffusive velocities υg and υv are related as

gas gas ρg υg + ρv υv = 0. (3.86)

In a similar way as it was done for the whole droplet, the vapor molar ﬂow towards the droplet and the change of the vapor molar mass of the droplet can be written as dr Φ = 4πr2ρgas d u υ (3.87) v d s,v dt − s − s,v and d 4 M˙ = πr3ρliq . (3.88) d,v dt 3 d d,v The amount of vapor molecules should be preserved, so M˙ = Φ , and after d,v − v some rearrangements an expression analogous to (3.83) can be written

dr r2ρgas(u + υ ) = ρliq ρgas r2 d . (3.89) − v v d,v − s,v d dt The rhs of the equation depends only on time, and it will be denoted as Fv:

dr F ρliq ρgas r2 d . (3.90) v ≡ d,v − s,v d dt

61 CHAPTER 3. NUCLEATION AND DROPLET GROWTH

Taking into account that the vapor molar fraction is

ρgas y = v , (3.91) v ρgas equation (3.89) can be rewritten in the following form, using (3.83) and (3.84):

F y F = r2ρgasυ . (3.92) v − v − v v

Because the molar fraction of the vapor component in the gas-vapor mixture is very small, Fick’s law determines the diffusion ﬂow of vapor

dy ρgasυ = ρgasD v , (3.93) v v − v dr where Dv is the diffusion coefﬁcient of the vapor component in the gas component.

After inserting (3.93) into (3.92), and integrating from(rd,ys,v) to(rfar,yfar,v), the expression for the droplet growth rate is:

liq gas 2 drd 1 1 ρd ρs rd = − dt rd − rfar (3.94) 1 y F/F ρgasD ln − far,v v . m m,v 1 y F/F − s,v v While integrating, it is assumed that the diffusion coefficient and gas molar density do not depend on the distance from the droplet, which is an approximation. There is a radial temperature field, which is caused by the release of latent heat. Because, the temperature gradient is not large, the integration can be done, but in the final expression the diffusion coefficient and molar density should to be taken at some intermediate temperature Tm between Td and Tfar. According to Hubbard [20] a one third rule is appropriate to apply

(2T + T ) T = d far . (3.95) m 3

From the deﬁnitions of F and Fv, it follows that

liq gas F ρd ρs = liq − gas , (3.96) Fv ρ ρs,v d,v − taking into account that the vapor molar fractions yfar,v, ys,v are very small and r r , equation (3.96) can be rewritten as d ≪ far 2 gas drd 2ρm Dm,v = liq gas (yfar,v ys,v) , (3.97) dt ρ ρs,v − d,v −

62 3.4. DROPLET GROWTH because ρliq ρgas equation can be further simpliﬁed, and the ﬁnal result is d,v ≫ s,v dr2 2ρgasD d = m m,v (y y ) , (3.98) dt liq far,v − s,v xd,vρd

liq liq where ρd,v was replaced with xd,vρd and xd,v is the molar fraction of vapor component in the droplet. In order to proceed further the heat ﬂow towards the droplet has to be determined.

Heat-transfer equation

The released latent heat flows into two directions: into the droplet, increasing the droplet temperature, and outward to the surrounding gas-vapor mixture, leading to a radial temperature field. The latent heat flow is indicated as Q˙ in Fig. 3.5. For the experimental conditions, the heat flow into the droplet is much smaller than the outward flow, as it was shown by Smolders [21]. Then the the conservation law for heat transfer can be written as

M˙ dL = Q,˙ (3.99) where L is the latent heat of condensation, which is the difference between molar enthalpies of the gas-vapor mixture and the liquid mixture in the droplet. The energy conservation equation (3.99), written without taking into account the heat absorbed by the droplet itself is called "wet-bulb" approximation.

For a steady state, the heat ﬂow Q˙ from the droplet to the gas-vapor mixture can be written as r r Q˙ = 4πk (T T ) far d , (3.100) m d − far r r far − d where km is the heat conductivity of the gas-vapor mixture at the intermediate temperature Tm. Combining this expression with (3.99) and taking into account relation r r yields d ≪ far gas L ρm Dm,v Td Tfar = (yfar,v ys,v) . (3.101) − km xd,v −

The temperature difference in the rhs of the equation can be estimated for the experimental conditions, and it is of the order of 0.1 K. At a pressure higher than 10 bar the gas-vapor mixture heat conductivity is high enough that the latent heat can be transferred to the surroundings without high temperature gradients. There- fore, the temperature difference between droplet and gas-vapor mixture is not taken into account and it is supposed that Td = Tfar. As a result, the subscript gas m in the droplet growth equation (3.98) can be replaced with "far " in ρm and

Dm,v, meaning that the molar density and diffusion coefﬁcient are taken at tem-

63 CHAPTER 3. NUCLEATION AND DROPLET GROWTH

perature Tfar.

Conditions at the droplet surface

The last quantities to discuss are the vapor molar fraction ys,v at the droplet surface and the vapor molar density xd,v in the droplet, which are both present in the growth rate equation (3.98). It is supposed that at the surface, the droplet eq and the gas-vapor mixture are in thermodynamic equilibrium, so ys,v = ys,v and eq xd,v = xd,v. The equilibrium compositions of the droplet and the gas-vapor mixture at the surface are deﬁned from the equalities of chemical potentials for the vapor and gas component in phase equilibrium:

liq eq gas eq µd,v Pd,xd,v = µs,v Pfar,ys,v , (3.102) liq eq gas eq µd,g Pd,xd,g = µs,g Pfar,ys,g . (3.103) Furthermore, the droplet and the gas-vapor mixture are in mechanical equilibrium.

The droplet pressure Pd and the pressure in the gas-vapor mixture Pfar are related by Laplace’s law: 2σ Pd = Pfar + . (3.104) rd For the smallest droplet r 0.15 µm observable in the experiments at 10 bar pres- d ≈ sure (at conditions for which the surface tension has the highest value σ 24 dyn/cm ≈ at 240 K), the Laplace pressure is about 3.2 bar.

The phase equilibrium equations (3.102-3.102) and the Laplace law contain temperature and pressure at far distance from the droplet. These far ﬁeld conditions can be identiﬁed with the temperature and pressure of the gas-vapor mixture con- stricted in the same volume as the real system, which includes the gas-vapor mixture and droplets. Apparently, this can be done, only if the volume occupied by the droplets is much smaller than the total volume of the system. This condition holds for the experiments, so the subscript far will be omitted from the equations. Then, for example, the conservation law for the vapor content in the system can be written as 4 y ρgas = n πr3ρliqx + ρgasy , (3.105) 0,v 0 3 d d d,v v gas where the y0,v and ρ0 are the initial molar fraction of vapor component and the molar density of the gas-vapor mixture. In this way the vapor depletion can be taken into account in the droplet growth calculation.

Ultimately, the driving force for droplet growth is the supersaturation of the gas- vapor mixture. The supersaturation can be deﬁned (A.15) as the ratio of the vapor eq molar fraction yv to the equilibrium vapor molar fraction yv . The droplet growth eq expression (3.98) includes the equilibrium molar fraction ys,v which differs from

64 3.4. DROPLET GROWTH

eq yv , because it is taken at equilibrium for a curved interface. To compare both molar fractions, the phase equilibria for the ﬂat and curved interfaces have to be considered; using the equality of the chemical potentials, the vapor component molar fractions in gas and liquid phases for both cases are related with the following expressions [22]:

P satφsat R T ln v v + R T ln(xeqγ ) = a yeqφ P a v v v v P (3.106) liq v (P ′) dP ′ − v sat PvZ and

sat sat Pv φv eq RaT ln eq + RaT ln xd,vγd,v = ys,vφs,vPfar Pd (3.107) liq v (P ′) dP ′. − v sat PvZ

For a small molar fraction of the vapor component, the vapor molecules interact only with the gas molecules, which means that the fugacity coefﬁcient does not depend on the molar vapor fraction and φv ≅ φs,v. Supposing that the pure vapor component in the liquid state is incompressible, subtracting (3.107) from (3.106) gives the expression which relates the equilibrium vapor molar fractions for the curved and ﬂat interfaces,

eq eq liq y x γd,v v (P P ) s,v = d,v exp v d − far . (3.108) yeq xeq γ R T far,v far,v far,v a far The exponential pre-factor in the equation is of the order of 1: for the ideal mixtures the activity coefﬁcients are close to unity, and a few bar pressure difference does not change the liquid composition appreciably. The exponent, say for methane/n- nonane mixture in the extreme case is as high as 1.3, which is certainly much less eq than the supersaturation ratio yv/yv . Therefore, it is a fairly good approximation eq eq to replace ys,v with yv in the droplet growth rate equation:

2 gas drd 2ρ Dv eq = eq liq (yv yv ) . (3.109) dt xv ρ −

The diffusion coefﬁcient of vapor molecules can be calculated with the Fuller expression [12].

65 CHAPTER 3. NUCLEATION AND DROPLET GROWTH

Comparison with experiment

In the experiments the droplet growth data are measured at known temperature and pressure. Equation (3.109) can be used in two ways: (1) the equilibrium vapor molar fraction and the diffusion coefficient can be determined by fitting experimental data, and (2) the mixture composition can be determined for known equilibrium molar fraction and diffusion coefficient.

In Fig. 3.6 a comparison of experimental data and model predictions is shown for a particular experiment. Droplet growth is followed for condensation in a methane/n-nonane mixture at 246.3 K and 37.6 bar. It is assumed that droplet growth obeys the diffusion-controlled law in the form (3.109) for very small droplets too. Experimental data and model predictions agree very well.

0.30 model exp

0.25

0.20 ) 2

m 0.15 µ ( 2 d r

0.10

0.05

0.00 0 5 10 15 20 25 30 35 t (ms)

Fig. 3.6: Example of n-nonane droplet growth in methane at 246.3 K and 37.6 bar.

2 The growth rate drd/dt in dependence of the vapor molar fraction can be determined from droplet growth data. The obtained result can be compared with the theoretical growth rate expression (3.109), as shown in Fig. 3.7. By fitting the experimental data with a straight line, the equilibrium vapor composition can be determined at droplet growth conditions from the extrapolated intercept with the yv-axes. The slope of the line yields the diffusion coefficient. So, it is possible to determined both the diffusion coefficient and equilibrium molar fraction, as it was demonstrated by Peeters et al. [14].

In many cases, however, the vapor molar fraction varies in a too narrow range, and an accurate determination of the equilibrium molar fraction appears impossible because of a too small range of yv. As was indicated by Peeters et al. [14], this difficulty can be overcome by adjusting the depth of the pressure pulse. A fit of the data with a straight line leads to large uncertainties in the diffusion coefficient

66 3.4. DROPLET GROWTH and equilibrium vapor molar fraction. Nevertheless, the method gives a unique possibility to determine experimentally the diffusion coefﬁcient and the equilibrium composition at conditions for which it is difﬁcult to do this with standard methods.

0.020

T = (242.1 +/- 0.6) K

p = (11.28 +/- 0.06) bar

0.015 /ms) 2 m

0.010

m /dt ( /dt 2

0.005 dr

0.000

0 5 10 15 20 25 30

y (x 10 )

Fig. 3.7: Surface growth rate example. Droplet growth data are obtained by Peeters [23] for methane/n-nonane mixture at 242 K and 11 bar. A closed square for zero surface growth rate indicates the n-nonane equilibrium molar fraction.

Another possibility to use droplet growth data is to check the quality of the prepared mixture. In the mixture the vapor molar fraction is very low, so it is difﬁcult to measure it directly. In an experimental run the mixture composition is determined by applying a suitable equation of state for the known conditions in the mixture preparation part of the experimental setup Sec. 2.5. The droplet growth data allows an independent check of the mixture composition: if the droplet growth data and model prediction coincide (as is shown in Fig. 3.7), then the mixture is prepared properly, and the obtained data is reliable.

The derived expression for diffusion-controlled droplet growth rate describes the droplet growth in mixtures with small vapor content and small amount of droplets at high pressure considerably well. Comparison of theory and experiment gives the possibility for an independent check of mixture composition and to determine some of the physical parameters.

67 CHAPTER 3. NUCLEATION AND DROPLET GROWTH

References

[1] F.F. Abraham, Homogeneous Nucleation Theory, Academic Press, New York, 1974.

[2] R.C. Tolman, J. Chem. Phys. 17, 333 (1949).

[3] K. Koga, X.C. Zeng, and A.K. Shchekin, J. Chem. Phys. 109 (1998).

[4] V.I. Kalikmanov, J. Chem. Phys. 124 (2006).

[5] R. Becker and W. Döring, Ann. Phys. 5, 719 (1935).

[6] G. Wilemski and B.E. Wyslouzil, J. Chem. Phys. 103, 1127 (1995).

[7] B. E. Wyslouzil, G. Wilemski, and R. Strey, in Nucleation and Atmospheric Aerosols 2000: 15th International Conference, pages pp. 724–727, 2000.

[8] D. Kashchiev, J. Chem. Phys. 76, 5098 (1982).

[9] D.W. Oxtoby and D. Kashchiev, J. Chem. Phys. 100, 7665 (1994).

[10] D.W. Oxtoby and A. Laaksonen, J. Chem. Phys. 102, 6846 (1995).

[11] C.C.M. Luijten, P. Peeters, and M.E.H. van Dongen, J. Chem. Phys. 111, 8535 (1999).

[12] R.C. Reid, J.M. Prausnitz, and B.E. Poling, The Properties of Gases and Liquids, McGraw–Hill Book Company, New York, 1987.

[13] M.J.E.H. Muitjens, Homogeneous condensation in a vapour/gas mixture at high pressures in an expansion cloud chamber, PhD thesis, Eindhoven University of Technology, 1996, ISBN 90-386-0199-9.

[14] P. Peeters, G. Pieterse, J. Hruby, and M.E.H. van Dongen, Phys. Fluids 16, 2567 (2004).

[15] J.C. Maxwell, Encyclopaedia Brittanica 2, 82 (1877).

[16] H. Hertz, Ann. Phys. 17, 177 (1882).

[17] M. Knudsen, Ann. Phys. 47, 697 (1915).

[18] G. Gyarmathy, The spherical droplet in gaseous carrier streams: review and synthesis, in Multiphase Science and Technology, volume 1, pages 99–279, Springer, Berlin, 1982.

68 REFERENCES

[19] J.B. Young, Int. J. Heat Mass Transfer 36, 2941 (1993).

[20] G.L. Hubbard, V.E. Denny, and A.F. Mills, Int. J. Heat Mass Transfer 18, 1003 (1975).

[21] H.J. Smolders, Non–Linear wave phenomena in a gas–vapour mixture with phase transition, Phd thesis, Eindhoven University of Technology, 1992.

[22] J.M. Prausnitz, R.N. Lichtenthaler, and E. Gomes de Azevedo, Molecular Ther- modynamics of Fluid-phase Equilibria, Prentice-Hall, New Jersey, 2nd edition, 1986.

[23] P. Peeters, Nucleation and condensation in gas-vapor mixtures of alkanes and water, PhD thesis, Eindhoven University of Technology, 2002.

69 CHAPTER 3. NUCLEATION AND DROPLET GROWTH

70 Chapter 4

Gradient-theory computation of the radius-dependent surface tension and nucleation rate for n-nonane clusters1

The classical theory of nucleation of droplets from supersaturated vapors has been elaborated by Becker and Döring [1] and others. One of the fundamentals of this theory is modeling the cluster as a droplet with a sharp phase interface. In reality the thickness of the phase interface is often comparable to the radius of the droplet. Possibly the cluster even may not possess a core identifiable with the bulk liquid phase. It is, therefore, more realistic to consider a diffuse interface model, in which the density varies continuously across the phase interface. Such model has been devised already by Van der Waals [2]. In this model, the local Helmholtz energy density not only depends on the local molar density and temperature, but also on the molar density gradient. Several interesting applications of the density gradient theory have been elaborated by Cahn and Hilliard [3]. The last authors also first applied the gradient theory to nucleation [4]. Theoretical foundations of the gradient theory and the related statistical physics background are given in the books by Rowlinson and Widom [5], Davis [6] and Kalikmanov [7]. A recent study of the gradient theory with comparison to molecular simulations was given by Baidakov et al. [8]. The gradient theory is closely related to the density functional theory, applied to gas-liquid nucleation by Oxtoby and Evans [9], nucleation in dipolar fluids by Talanquer and Oxtoby [10], nucleation of non-ideal binary mixtures by

1This chapter is the text of a manuscript in preparation with authors: J. Hrubý, D.G. Labetski, and M.E.H. van Dongen CHAPTER 4. GRADIENT-THEORY COMPUTATION OF SURFACE TENSION AND NUCLEATION RATE FOR N-NONANE CLUSTERS

Laaksonen and Oxtoby [11] and to nucleation of amphiphilic binary mixtures by Napari et al. [12, 14]. Both theories are based on the mean-ﬁeld approximation. A novel nucleation theory based on mean-ﬁeld considerations has recently been proposed by Kalikmanov [13].

4.1 Theory

Gibbsian droplet model

One assumes that the interior of the droplet is formed by a liquid at pressure pL, which is higher than the pressure pG of the surrounding gas phase due to the Laplace pressure

2σ pL pG ∆p = , (4.1) − ≡ rs

where σ is the surface tension and rs is the radius of the droplet. More accurately, rs is the radius of the sphere of tension [5]. Only for this radius deﬁnition the Laplace equation can be written in the simple form of Eq. (4.1). Of special importance in the classical nucleation theory is the critical cluster. The radius of the critical cluster is such that the probabilities of growing and shrinking are equal. In other words this is a condition of zero net ﬂux. In thermodynamic language this translates to the equality of chemical potentials of the liquid and gas phases,

µL(pL,T ) = µG(pG,T ) . (4.2)

When an equation of state is available, Eq. (4.2) can be used to compute the liquid pressure and the pressure difference ∆p for the given gas pressure. Using an equation of state valid for bulk liquid may be in error if the cluster is so small that the thickness of the phase interface is similar to the cluster radius. However, it is practical to use the pressure difference computed from Eq. (4.2) as a measure of supersaturation. If the surface tension is known, the work of formation ∆Ω of the critical cluster can be computed as

1 ∆Ω = σA , (4.3) 3 s

2 where As = 4πrs now is the surface area of the (critical) droplet. An inherent part of the classical nucleation theory is the so-called capillarity approximation: we approximate the actual surface tension of the cluster with the surface tension of the ﬂat phase interface at thermodynamic equilibrium, a quantity which can be found in tables. Accepting the capillarity approximation, the set of equations

72 4.1. THEORY

Eq. (4.1) to Eq. (4.3) is complete and we can compute the work of formation.

In the present work we go beyond the capillarity approximation. Using the gradient theory we will directly compute the work of formation ∆Ω for given conditions of the gas phase. Then, using Eqs. (4.1) and (4.3), we compute the corresponding radius and surface tension,

3∆Ω 1/3 r = , (4.4) s 2π∆p

3∆Ω 1/3 σ = . (4.5) 16π∆p2

The gradient theory also allows computation of the excess number of molecules ∆N. In the Gibbsian picture the excess number of molecules is given as

∆N = Vs∆ρ + AsΓ , (4.6) where V = 4 πr3 is the volume of the sphere of tension, ∆ρ ρ ρ , and Γ is the s 3 s ≡ L − G surface excess of molecules or adsorption. Equation (4.6) will be used to extract the adsorption Γ from gradient theory computations. From the differential analysis of the equations deﬁning the droplet model it follows that the surface tension and adsorption are related by the Gibbs adsorption equation [5]:

dσ = Γ dµ . (4.7) −

This equation is valid for isothermal changes of the critical droplet, Eq. (4.2) is satisﬁed along the path of differentiation.

Alternatively, the droplet can be modeled as a discontinuous proﬁle without surface excess, thus deﬁning the so-called equimolar surface and equimolar radius re: 4 ∆N = πr3∆ρ . (4.8) 3 e

The difference of the two radii is the Tolman [15] length,

δ r r . (4.9) ≡ e − s

The importance of the Tolman length is in its relation to the dependence of the surface tension on the radius. In the present work we discuss rather the dependence on the pressure difference ∆p. [Both are related by Eq. 4.1.] Using the isothermal Gibbs-Duhem equations for both phases we derive

73 CHAPTER 4. GRADIENT-THEORY COMPUTATION OF SURFACE TENSION AND NUCLEATION RATE FOR N-NONANE CLUSTERS

d∆p dµ = . (4.10) ∆ρ

Substituting this expression to the Gibbs adsorption equation (4.7) we ﬁnd

dσ = t d∆p , (4.11) − where

t Γ/∆ρ (4.12) ≡ has a dimension of length. The lengths δ and t can be related by comparing Eqs. (4.6) and (4.8):

δ2 δ3 t = δ + + 2 . (4.13) rs 3rs

For the planar interface (rs ) these lengths are equal: t = δ . → ∞ ∞ ∞

Gradient theory

The gradient theory operates with continuous density proﬁles ρ(r). The local Helmholtz energy density of an inhomogeneous ﬂuid comprises two terms:

1 dρ 2 Φ = Φ (ρ) + c . (4.14) hom 2 dr

The first term is the Helmholtz energy density of a homogeneous fluid at the actual local density. The second term is a correction for inhomogeneity. This is represented by the gradient square term and a positive coefficient c, called the influence parameter. In this work the influence parameter is independent of density. The influence parameter can be related to the direct correlation function of the homogeneous fluid. This approach is not practical for non-spherical molecules. Rather the influence parameter is obtained by matching the experimental value of the surface tension of the flat phase interface as discussed later. The chemical potential is homogeneous throughout the system. Therefore, the grand potential density is given as

ω = Φ ρµ . (4.15) − G

In a homogeneous system, the grand potential density reduces to p . The work − G of formation of a cluster can be expressed as a difference between the grand potential of a system containing the cluster and the gas phase and the grand potential of

74 4.1. THEORY the system containing the gas phase only:

2 ∞ 1 dρ ∆Ω = ∆ω (ρ) + c 4πr2 dr . (4.16) hom 2 dr Z0 " # where the excess grand potential density ∆ωhom is given as

∆ω (ρ) Φ (ρ) ρµ + p . (4.17) hom ≡ hom − G G

The function ∆ωhom(ρ) is obtained from a suitable equation of state of the ﬂuid. The surface tension of the ﬂat interface can be obtained in a very similar fashion:

2 ∞ 1 dρ σ = ∆ωhom(ρ) + c dz , (4.18) ∞ 2 dz Z−∞ " # where z is a coordinate perpendicular to the phase interface. To be able to compute the work of formation of a cluster using Eq. (4.17), the density profile ρ(r) is required. Similarly, the density profile ρ(z) is needed to compute the surface tension of the flat phase interface.

For the flat surface, the density profile is found by utilizing the general principle that a thermodynamic system of given volume, temperature, and chemical potential will equilibrate in a state of lowest grand potential. The task is to find a density profile minimizing integral (4.18). The necessary condition is that the profile is sta- tionary, i.e. for small perturbations around this profile the value of the integral will not change. This functional problem can be reduced to a second-order Euler- Lagrange equation

d2ρ 1 = ∆µ(ρ) , (4.19) dz2 c where ∂∆ω ∆µ hom = µ(ρ) µ . (4.20) ≡ ∂ρ − G

Eq. (4.19) is to be solved for boundary conditions ρ( ) = ρ , ρ( ) = ρ , where −∞ L ∞ G ρL and ρG are densities of the homogeneous liquid (L) and gaseous (G) phases in thermodynamic equilibrium, determined by the equating the pressures and chemical potentials. First, the integral for Eq. (4.19) can easily be found yielding expressions for the density proﬁle and surface tension:

ρ(z) c 1/2 z = dρ , (4.21) 2∆ω (ρ) Zρ(0) hom

75 CHAPTER 4. GRADIENT-THEORY COMPUTATION OF SURFACE TENSION AND NUCLEATION RATE FOR N-NONANE CLUSTERS

ρL 1/2 1/2 σ = (2c) [∆ωhom(ρ)] dρ . (4.22) ∞ ZρG

The density profile ρ(r) of the critical cluster will not minimize the grand potential, because the critical cluster is in labile (not stable) equilibrium with the gas phase. The critical density profile has rather a nature of a saddle point. Roughly speaking, it is a minimum of the grand potential with respect to variations of the shape of the cluster, and a maximum with respect to variations of its radius. This problem has been briefly discussed by Oxtoby and Evans [9] for the case of density functional computations. Contrary to the density functional theory, in the gradient theory it is possible to solve both the planar and the spherical cases by Euler-Lagrange equations. For the critical cluster, the Euler-Lagrange equation corresponding to the functional (4.16) is

d2ρ 2 dρ 1 + = ∆µ(ρ) . (4.23) dr2 r dr c

Equation (4.23) must be solved numerically for boundary conditions ρ(r ) = → ∞ ρG and ( dρ/ dr)(r = 0) = 0.

The above given equations are sufficient for the computation of the density profiles, the planar surface tension, and the work of formation of clusters. In addition we mention formulas which proved useful in assessing the numerical accuracy of the computed density profiles. Let us assume that a profile ρ(z) satisfies the Euler- Lagrange equation (4.19). We use the chain rule ( d∆ω / dz) = ∆µ ( dρ/ dz) and hom · for ∆µ we substitute from Eq. (4.19). By integration we obtain

2 ∞ dρ σ = c dz . (4.24) ∞ dz Z−∞

Using this result, the density gradient can be eliminated from Eq. (4.22). In this way we obtain

∞ σ = 2 ∆ωhom dz . (4.25) ∞ Z−∞

In Eq. (4.24) the surface tension only depends on the density gradient. On the other hand, in Eq. (4.25) the surface tension is given in terms of the density profile only. Comparison of the planar surface tensions computed using these two formulas is an efficient test of the numerical accuracy of the computed profile ρ(z).

In a similar fashion, we derived alternative formulas for the computation of the work of formation of the critical cluster:

2 1 ∞ dρ ∆Ω = c 4πr2 dr , (4.26) 3 dr Z0 76 4.2. SURFACE TENSION AND THE WORK OF FORMATION

∞ ∆Ω = 2 ∆ω 4πr2 dr . (4.27) − hom Z0

Another important output of the gradient theory computations is the excess number of molecules. For the continuous proﬁle it is computed as

∞ ∆N = [ρ(r) ρ ] 4πr2 dr . (4.28) − G Z0

In the limit r (planar phase interface) the excess number of molecules s → ∞ Eq. (4.28) becomes inﬁnite. The surface excess can be expressed in terms of the Tolman length

δ = ze zs , (4.29) ∞ −

where ze is the position of the equimolar surface, and zs is the position of the surface of tension. Provided that the liquid phase exists at z and the gas → −∞ phase exists at z , the equimolar surface is at a position with respect to an → ∞ arbitrary coordinate system

0 1 ∞ ze = [ρ(z) ρL] dz + [ρ(z) ρG] dz . (4.30) ρL ρG − 0 − − Z−∞ Z

The position of the surface of tension with respect to the same coordinate system is computed based on the ﬁrst moment of the gradient square term:

2 1 ∞ dρ z = c z dz . (4.31) s σ dz ∞ Z−∞

4.2 Surface tension and the work of formation

The gradient-theory computations were based on the Peng-Robinson equation of state [37]. The influence parameter was obtained by fitting the orthobaric surface tension data for n-nonane by Jasper et al. [16]. The influence parameter was computed using Eq. (4.22) valid for the planar phase interface. The experimental surface tension data were represented within their accuracy using a constant 66 5 (temperature-independent) influence parameter c = 3.0308 10− J m . ×

The gradient theory computations of critical clusters were performed at 230 K. This isotherm was chosen because of the availability of experimental nucleation rate data. The corresponding vapor pressure computed from the Peng-Robinson equation of state was psat,PR = 3.20449 Pa. We computed 69 density proﬁles for vapor pressures ranging form 8 p to 33000 p . The proﬁles are shown × sat,PR × PR 77 CHAPTER 4. GRADIENT-THEORY COMPUTATION OF SURFACE TENSION AND NUCLEATION RATE FOR N-NONANE CLUSTERS

4 ) 3

3 (kmol/m ρ

0 0 0.5 1 1.5 2 2.5 3 3.5 4 r (nm)

Fig. 4.1: Density profiles of n-nonane clusters at 230 K computed with the gradient theory. Profiles are computed for vapor pressure multiples pG/psat,PR =8, 11, 15, 22, 30, 40, 50, 70, 100, 140, 200, 300, 400, 500, 600, 800, 1000, 1200, 1500, 2000,...,19000, 19500, 20000, 21000,...,32000, 33000. in figure 4.1. For pressures smaller than 8 p the iteration did not reach our × sat,PR convergence criterion (discussed below). From above the pressure range was limited by the vapor-liquid spinodal at 107951 Pa. The computation has been carried out using the BVP4 solver in MATLAB.

Figure 4.2 shows the works of formation computed as an arithmetic average of formulas (4.26) and (4.27).

Figure 4.3 shows the excess numbers of molecules, computed using Eq. (4.28). ∆N is a monotonically decreasing function of ∆p except for very close to the spinodal where it reaches a minimum and increases again, finally diverging at ∆p = 1081.4 bar where the spinodal is located. Close to the spinodal, a similarity solution of the Euler-Lagrange equation (4.23) can be developed as shown by Cahn and Hilliard [4] and in more detail by Wilemski and Li [17]. This is a region of diffuse density fluctuations. The resulting profiles (see figure 4.1) are rather an artefact of the mean field approximation. The clusters relevant to nucleation (at least in this study) are on the decreasing branch of the ∆N curve.

Figure 4.4 shows the surface tension computed using Eq. (4.5) from the previously determined works of formation Eq. (4.26) and Eq. (4.27). The difference of the surface tensions computed with these formulas was used as a necessary condition for

78 4.2. SURFACE TENSION AND THE WORK OF FORMATION

3 10

2 10

1 10 kT / ∆Ω 0 10

−1 10

−2 10 0 200 400 600 800 1000 1200 ∆ p (bar)

Fig. 4.2: The work of formation ∆Ω of n-nonane clusters at 230 K as function of the theoretical pressure difference ∆p.

5 10

4 10

3 10 N ∆

2 10

1 10

0 10 0 200 400 600 800 1000 1200 ∆ p (bar)

Fig. 4.3: Excess number of molecules ∆N of n-nonane clusters at 230 K as function of the theoretical pressure difference ∆p.

79 CHAPTER 4. GRADIENT-THEORY COMPUTATION OF SURFACE TENSION AND NUCLEATION RATE FOR N-NONANE CLUSTERS

15 (mN/m) σ

0 0 200 400 600 800 1000 1200 ∆ p (bar)

Fig. 4.4: Surface tension of n-nonane clusters at 230 K as function of the theoretical pressure difference ∆p.

terminating the iterative computation of the density profiles. It was required that 6 the relative difference is smaller than 1 ppm (1 10− ). Also shown in figure 4.4 is × the surface tension for the planar phase interface (diamond at ∆p = 0) computed with Eq. (4.22). With increasing ∆p the surface tension initially slightly increases, reaches a maximum and then decreases along a line appoximately according to a quadrant of a circle, finally vanishing at the spinodal.

The solid line in Figure 4.4 is obtained by a piece-wise cubic interpolation between these points. This line is a piece-wise cubic

2 3 fi(∆p) = h0i + h1i∆p + h2i(∆p) + h3i(∆p) , (4.32)

with coefﬁcients h0i,...,h3i obtained form the surface tensions and lengths t in the boundary points, using Eq. (4.11:

f (∆p ) = σ , f ′(∆p ) = t , f (∆p ) = σ , f ′(∆p ) = t . (4.33) i i i i i − i i i+1 i+1 i i+1 − i+1

For the left conditions of the first interval we use the surface tension σ and Tol- ∞ man length δ for the plane phase interface computed using Eqs. (4.22) and (4.29), ∞ respectively. A great advantage of Eq. (4.33) is that it enables an accurate interpolation between the planar interface and the first computed cluster. The range of very large clusters which are difficult for direct computation is bridged. Another ad-

80 4.3. NUCLEATION RATE

0.4

0.35

0.3

0.25

0.2 nm

0.15

0.1 δ=r −r e s 0.05 t=δ+δ2/r +δ3/3r2 s s 0 δ = t ∞ ∞ −0.05 0 200 400 600 800 1000 1200 ∆ p (bar)

Fig. 4.5: Tolman length δ of n-nonane clusters at 230 K as function of the theoretical pressure difference ∆p. vantage is in checking the numerical consistency of the computations. The values (surface tensions) are computed from the works of formation ∆Ω solely, whereas the slopes are computed based on the excess numbers of molecules ∆N. If the computed density profiles were inaccurate, the resulting line would be wavy. A second test of the consistency is a plot of the Tolman length δ and the length t as functions of ∆p, shown in figure 4.5. Here the circles are the Tolman lengths computed with Eq. (4.9) and Eq. (4.29) at ∆p = 0 and the x-symbols are the lengths t computed using Eq. (4.13). The solid line, interconnecting the t-values is a piece-wise quadratic curve. From this line, by solving Eq. (4.13), the solid line interconnecting the Tol- man lengths (circles) is obtained. As shown in figure 4.5, the lengths δ and t are initially practically identical and are represented by an almost linear function of ∆p. The quadratic term is insignificant for large clusters. Unlike the original result by Tolman [15], the Tolman length is initially negative, corresponding to an increase in surface tension. Both lengths turn positive at ∆p 200 bar. For smaller ≈ clusters (large ∆p) the length t diverges faster. On the other hand, the behavior of the Tolman length δ is more complex, showing two inversion points.

4.3 Nucleation rate

The gradient theory in the present form only enables computation of critical clusters. However, the works of formation for non-critical clusters can be estimated

81 CHAPTER 4. GRADIENT-THEORY COMPUTATION OF SURFACE TENSION AND NUCLEATION RATE FOR N-NONANE CLUSTERS with a good accuracy. Perhaps the most general approach to this problem available today is based on the “non-critical nucleation theorem”, derived by Hrubý [18] and Bowles et al. [19]. We assume that the cluster is deﬁned by some microscopic con- straint. Here the number of molecules forming the cluster,N, is taken constant. When the condition of the vapor changes at constant temperature and constant N, the theorem says that the work of formation is changed as

d(∆Ω ) = ∆N dµ . (4.34) N − N G

From this relation, the classical nucleation theorem [20] can be derived. Here another aspect is important. By integrating at constant N, changing the condition of the vapor isothermally between the chemical potential µN∗ where the cluster of N molecules is critical and the actual chemical potential of the gas phase µG we obtain:

µG ∆ΩN (µG) = ∆ΩN (µN∗ ) ∆NN dµ . (4.35) − µ∗ Z N

Here, ∆ΩN (µN∗ ) is the work of formation of the cluster of N molecules computed with the gradient theory, and ∆ΩN (µG) is the work of formation of the non-critical cluster. It is always possible to express the excess number of molecules as

∆N = N V ρ , (4.36) N − N G

where VN represents the volume of the cluster of N molecules. The exact value of VN is, in fact, deﬁned by Eq. (4.36). Then we can proceed with integrating Eq. (4.35):

µG ∆ΩN (µG) = ∆ΩN (µN∗ ) N (µG µN∗ ) + VN ρG dµ . (4.37) − · − µ∗ Z N

Making an approximation (the only approximation in the derivation) that VN is independent of the condition of the gas, we ﬁnd

∆Ω (µ ) = ∆Ω (µ∗ ) N (µ µ∗ ) + V (p p∗ ) , (4.38) N G N N − · G − N N · G − N

where pN∗ is the pressure of the gas phase at the condition when the cluster of N molecules is critical.

The number density of non-critical clusters of N molecules is then obtained as

1 ∆Ω (µ ) n = exp N G . (4.39) N ϑ − k T B

82 4.4. COMPARISONS

In this equation, the pre-factor 1/ϑ depends on the form of the nucleation theory.

For the present computations we used the classical choice of 1/ϑ = n1. As a next step we determine the impingement rate CN of vapor molecules on the surface of cluster of N molecules:

k T C = A n B . (4.40) N N 1 2πM r v

Here AN is the surface area of the cluster, n1 is number density of free vapor molecules (monomers), and Mv is molecular weight of the vapor. By solving the nucleation kinetics in the steady state we obtain the nucleation rate

1 Nmax 1 − J = . (4.41) CN nN ! N=XNmin

Theoretically the summation should run from 1 to . The actual minimum and ∞ maximum cluster sizes were taken as Nmin = 2 and Nmax = 800. The computed nucleation rates were insensitive to variation of these bounds, because the terms of Eq. (4.41), corresponding to clusters much smaller or bigger than the critical cluster are small.

The computation was organized as follows. With help of the interpolating func- 3 tion (4.32) a long table was prepared containing columns 1/N , 1/rs, 1/re, σ, µN∗ , ln pN∗ . The rows in this table corresponded to non-integer values of N. In this table we interpolated to obtain the values for integer N’s. The work of formation was computed as

1 2 4 3 ∆Ω (µ ) = σ4πr N (µ µ∗ ) + πr (p p∗ ) . (4.42) N G 3 s − · G − N 3 e · G − N

4.4 Comparisons

Nucleation rate

The nucleation rate of n-nonane in helium has been measured by Hung et al. [21, 5 3 1 22] (233 to 313 K, 5 10− to 1 cm− s− ) and by Rudek et al. [23] (257 to 313 K, × 4 3 1 5 10− to 5 cm− s− ) using an upward diffusion cloud chamber. The two data × sets are consistent. Measurements by Rudek et al. are not considered here because they are at higher temperatures. Expansion-chamber measurements have been reported by Adams et al. [24], Wagner and Strey [25], and Viisanen et al. [26] (in 5 8 3 1 argon at 230 K, 0.5 bar, 4 10 to 2 10 cm− s− . Of the expansion chamber ≈ × × measurements, only data by Viisanen et al. has been considered, because of su-

83 CHAPTER 4. GRADIENT-THEORY COMPUTATION OF SURFACE TENSION AND NUCLEATION RATE FOR N-NONANE CLUSTERS perior quality. Expansion wave tube experiments have been reported by Luijten et al. [27, 28]. The data by Luijten et al. are at higher pressures: nominally 10, 25, and 40 bar. Only the 10 bar measurements were used for the present comparisons.

For the expansion chamber and expansion wave tube experiments we re-evaluated the experimental supersaturations starting from the experimental temperatures, pressures, and molar fractions. We used the saturated vapor pressure equation by King and Al-Najjar [29], consistently with the diffusion cloud chamber data by Hung et al. [21]. We also considered the real-gas effects on supersaturation. The supersaturations were determined using deﬁnition

µ µ S = exp v − v,sat , (4.43) k T B where µv is the chemical potential of the vapor (n-nonane) in the nucleation condition and µv,sat is the chemical potential of the vapor at phase equilibrium with liquid, in the presence of the background gas (He or Ar) at the same chemical potential µg as in the nucleation condition. For the gas phase we used the virial equation of state truncated after the second virial coefficient. For the argon-argon virial coefficient we used the correlation by Bokis et al. [30]. The argon-nonane virial coefficient was computed according to Kaul et al.. The second virial coefficients for helium-helium and helium-nonane were neglected. The supersaturation (4.43) also contains the Pointing effect due to the pressure of the background gas. As discussed by Fisk and Katz [31], the Pointing effect influences both the experimental supersaturations and the nucleation theory in a very similar fashion. In order to make possible direct comparison of nucleation data obtained at different background pressures, we defined a supersaturation reduced to pure vapor condition as p p S = S exp − v,sat . (4.44) 0 k T ρ B L

The experimental data is at temperatures slightly deviating (up to 4 K) from the

Tnom = 230 K isotherm. We adjusted the supersaturations further to compensate for this:

SICCT(Jexper,Tnom) Scorr = S0 , (4.45) SICCT(Jexper,Texper)

where SICCT(J, T ) is the inverse function to the so-called internally consistent classical nucleation theory as given by Girshick and Chiu [32]. The experimental nucleation rate data shown in ﬁgure 4.6 are corrected according to Eq. (4.45).

Figure 4.6 also shows the theoretical line computed with the gradient theory. This

84 4.4. COMPARISONS

15 10

) 10 10 −1 s −3 (m J

5 10 Viisanen et al. 1998 Luijten et al. 1999 Hung et al. 1989 CNT ICCT Gradient Theory 0 10 30 50 100 200 500 S

Fig. 4.6: Nucleation rate J of n-nonane clusters at 230 K as function of supersaturation S.

line was computed for given supersaturations using Eq. (4.41). Also shown is the classical nucleation theory by Becker and Döring and the internally consistent theory by Girshick and Chiu. A problem is that the gradient theory computations are done with the Peng-Robinson equation of state, which yields considerably inaccurate vapor pressure and liquid density for n-nonane at 230 K. To make possible the comparison of the theories, we computed the classical and internally consistent theories with the liquid density and vapor pressure given by the Peng-Robinson equation of state. The comparison with the classical theory is then quite clear: At low supersaturations the gradient theory gives lower nucleation rates than the classical theory. This is due to the fact that for the large critical clusters the surface tension obtained with the gradient theory (ﬁgure 4.4) is higher than the planar surface tension used in the classical theory. For higher supersaturations, the critical and near-critical clusters playing the decisive role in the nucleation kinetics get smaller. According to ﬁgure 4.4 the corresponding surface tension gets smaller than at the planar phase interface and the nucleation rate predicted by the gradient theory gets higher than for the classical nucleation theory. As result, the gradient theory computations predict a steeper log J vs. log S line. The internally consistent theory shows a relatively good agreement with the experimental data. This is mainly due to a different pre-factor in formula (4.39) and the agreement can be considered as rather fortuitous.

85 CHAPTER 4. GRADIENT-THEORY COMPUTATION OF SURFACE TENSION AND NUCLEATION RATE FOR N-NONANE CLUSTERS

4.5 Conclusions

We performed gradient-theory computations of the density profiles, works of formation, and excess numbers of molecules of n-nonane clusters. These results were interpreted in the framework of the Gibbsian droplet model with variable surface tension. This interpretation allowed construction of a very efficient interpolating scheme and a check of numerical accuracy of the computations. We also developed a way of estimating the works of formation of non-critical cluster in order that the complete nucleation kinetics can be evaluated. Finally, we computed the nucleation rate by adaptation of the standard steady state nucleation kinetics and compared the theoretical prediction to the experimental data. The prediction is not yet fully quantitative for several reasons. First, the performance of the Peng-Robinson equation of state is poor. The saturated vapor pressure and liquid density computed from the Peng-Robinson equation for n-nonane at 230 K exhibit large errors. This is a problem which can be improved relatively easily. Second, the problem of variability of the surface tension is not the only unresolved problem of the nucleation theory. Another problem is associated with the pre-factor of Eq. (4.39), which was discussed recently in terms of the so-called translational volume scale ϑ. Upon the condition that a sufficiently accurate mean-field equation of state is available, the gradient theory is with certainty superior to the capillarity approximation, assuming a constant surface tension. On the other had, it is to be realized that the gradient theory is based on some severe approximations. It is to be expected that the gradient theory performs well if the gradient length of the density profile is large in comparison with the range of the intermolecular forces. For the examples considered in this paper, this condition is only marginally met.

4.6 Thermophysical properties of n-nonane

Surface tension

The surface tension for n-nonane has been measured by Jasper and co-workers 2 [16, 33]. Orthobaric data [16] span over the range 0–60◦C and has been correlated by the authors to a linear form

σ = 24.84 0.09417 (T 273.15) . (4.46) − −

Isobaric measurements [33] were taken in a dry nitrogen atmosphere at a total pressure of 760 Torr (101325 Pa) in the range of 0–100◦C. The data has been correlated by the authors to

2Orthobaric: measured at the vapor pressure of the pure n-nonane.

86 REFERENCES

σ = 24.72 0.09347 (T 273.15) . (4.47) − −

Adsorbed nitrogen decreases the surface tension. This effect is more pronounced at lower temperatures because the gas molecules with lower thermal energy adsorb more easily. Consequently, both the surface tension at 273.15 K and the slope have somewhat lower values for the isobaric case than for the orthobaric case.

Vapor pressure

The vapor pressure of saturated n-nonane has been discussed in detail by Hung et al. [21]. These authors recommended equation by King and Al-Najjar [29], al- though a different correlation was used in a later study by the J. Katz group [23]. More recently, a generalized vapor pressure equation has been obtained by Lem- mon and Goodwin [34] (their Eq. 12). These authors also provide accurate corre- lations of the critical temperature, critical pressure, and the accentric factor (their equations 20, 21, and 24, respectively). We adopted these values. For the vapor pressure we used the King and Al-Najjar correlation.

Liquid density

Density of the liquid n-nonane was computed from relations given by Assael et al.[35]. This correlation is based on experimental data in temperature range of 303 to 423 K. Fundamental equation of state by Lemmon and Span [36] provides a superior representation of liquid density and compressibility. Concerning the vapor pressure, however, it is not supported by experimental direct experimental data. Also data for liquid and gas heat capacities are missing

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87 CHAPTER 4. GRADIENT-THEORY COMPUTATION OF SURFACE TENSION AND NUCLEATION RATE FOR N-NONANE CLUSTERS

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89 CHAPTER 4. GRADIENT-THEORY COMPUTATION OF SURFACE TENSION AND NUCLEATION RATE FOR N-NONANE CLUSTERS

90 Chapter 5

Results and discussion

5.1 n-Nonane nucleation in methane

In the methane/n-nonane mixture studied, methane is a non-condensable component. Its critical temperature is lower than the condensation temperature in the experiments. Methane serves as a carrier gas, it consumes the latent heat of vapor condensation, and so it holds the nucleation temperature constant. It is expected that methane also influences the nucleation process. Two different mechanisms of influence can be distinguished. First methane changes the partial saturation pressure of n-nonane and so its supersaturation: the saturation pressure of pure n-nonane differs from the partial saturation pressure of n-nonane in a methane/n- nonane system. Second, methane itself influences the nucleation process. The phase equilibrium calculation shows that the molar fraction of methane in the liquid phase at 40 bar and 240 K is approximately 0.2. It can be expected that the methane concentration in the critical cluster will be of the same order of magnitude; as a result, the surface tension of a critical cluster in the methane/n-nonane mixture will differ from the surface tension of a pure n-nonane cluster. Further- more, at the experimental conditions the critical clusters are very small – tens of molecules. The composition of the surface layer will differ from the composition of the surface layer of the flat interface because of the curvature effect. The surface tension of the cluster will be different from the surface tension of a flat interface. The purpose of the experiments is to obtain data which can help to form a better insight into the mechanism of the influence of methane on nucleation of n-nonane.

The ﬁrst series of high pressure experiments on nucleation of n-nonane in methane has been carried out with our setup by Luijten et al. [1]. They obtained n-nonane nucleation rate data at pressures of 10, 25, 40 bar and at a constant temperature of CHAPTER 5. RESULTS AND DISCUSSION

240 K. Later the experimental setup was improved and more accurate data was obtained by Peeters et al. [2] at 40 bar. It was observed that n-nonane nucleation is strongly inﬂuenced by the presence of methane. The values of the nucleation rates are higher at higher carrier gas pressures for the same n-nonane supersaturation. The striking feature of these experiments was a decrease of the slope of the nucleation isotherms as a function of supersaturation with increasing pressure, in particular between 25 and 40 bar. It is emphasized again that the slope of the nucleation isotherm gives the number of n-nonane molecules in the critical cluster (Sec. 3.3). A similar change of the slope between 10 and 40 bar is not observed for experiments with nitrogen as a carrier gas. The explanation is that the solubility of nitrogen in liquid n-nonane is much less than the methane solubility. To investigate further how the slope of the nucleation isotherm is inﬂuenced by pressure an additional series of experiments has been performed. The additional data also gives the possibility to estimate the critical cluster composition with more accuracy than it has been done so far. The additional series is obtained with the same experimental setup, but there are some changes in the experimental procedure. In this study in situ calibration of pressure was applied for the dynamic pressure transducer, as explained in Sec. 2.3. In Peeters’ experiments the dynamic pressure transducers were calibrated prior to an experimental series, and the reading from the dynamic pressure transducer is used to deduce pressure.

Experimental data

The present experimental setup allows to measure nucleation rates in a window 13 18 1 3 from 10 to 10 s− m− . In order to keep nucleation rates in this range when the pressure increases, the n-nonane supersaturation should be decreased. The decrease of supersaturation is achieved by decreasing the molar fraction of n-nonane in the initial mixture. For 40 bar experiments the molar fraction of n-nonane is in the range of 50-100 ppm, and for 10 bar experiments the molar fraction is in the range of 200-300 ppm. This already indicates the inﬂuence of methane on nucleation: to hold the nucleation rate in the same range, the n-nonane molar fraction has to be shifted to lower values with pressure increase in order to keep the supersaturation constant. This is a consequence of the real-gas effect (vapor pressure enhancement) and partly explains the effect of methane on nucleation. Here, the supersaturation is calculated for given temperature and pressure with the RKS- EOS using the ﬂash calculation for the initial mixture composition as a feed. The initial mixture compositions are calculated for known saturation conditions with the equations from Sec. 2.5, the data needed to calculate the initial mixture composition is listed in Appendix C.

The obtained data are collected in the corresponding tables in Appendix C. Nu- cleation experiments are performed at about 33.5 bar and 240 K. The results of

92 5.1. N-NONANE NUCLEATION IN METHANE new experiments together with the old data are presented in Fig. 5.1. The nucleation rate data are shown as functions of n-nonane supersaturation. The scattering observed in the data is partly caused by variation in the conditions at which the nucleation rates were measured. The pressures and temperatures for different data points are not exactly the same, they vary from one experiment to another in the following ranges: 0.2 bar and 0.5 K. Similar variations of temperature and pres- ± ± sure pertain to the Peeters and Luijten data. The new data, marked as the closed circles in the ﬁgure, is consistent with the earlier obtained data: the new nucleation isotherm is situated between the 40 bar and 25 bar nucleation isotherms. To guide the eye a straight line is drawn through the data points. It is seen that the slope of the nucleation isotherms is gradually decreasing with pressure. The slope of the new isotherm is in between the slope of the 40 and 25 bar isotherms.

1018

1017

16 ) 10 -1 s -3

J (m 1015

1014

1013 2.5 5 10 20 40 80

Sc9 40 bar, Peeters 25 bar, Luijten 33 bar, Labetski 10 bar, Luijten

Fig. 5.1: Nucleation rate of n-nonane in methane, as a function of n-nonane supersaturation for different pressures. For all experiments the average temperature is 240 K.

The experimental data can be compared with the quasi-unary nucleation theory. It is based on a unary nucleation model with an effective medium having the properties of the two-phase gas-vapor mixture (see Sec. 3.2). The predictions of the quasi-unary nucleation theory are indicted in Fig. 5.2 as dashed lines. For low pressure the theory and data are not far apart. The difference can be attributed to uncertainties in the equilibrium state and therefore to uncertainties in the supersaturations. But, with the pressure increasing, the theory and experiment are strongly deviating. It should be noted, that the quasi-unary nucleation theory predicts an inﬂuence of methane on nucleation: for the same supersaturations of n- nonane the nucleation rates are higher at higher pressures. This trend agrees with the experimental observations. One of the important characteristics of the effective medium (see Sec. 3.2) is the surface tension. For the surface tension, the expression suggested by Luijten and van Dongen [3] for a ﬂat interface is employed, which means that the effect of the surface curvature is not taken into account. The expres-

93 CHAPTER 5. RESULTS AND DISCUSSION sion is based on the experimental data by Deam and Maddox [4] for a two-phase methane/n-nonane system at equilibrium. So, the change of surface tension with pressure is taken into account. What is also important is that the quasi-unary theory does not predict the change of the slope of the nucleation isotherms with pressure. The theory even predicts a slight increase in the slope with pressure, opposite to what is observed. Because the quasi-unary theory is based on the equilibrium properties of the two-phase system, the properties of the critical cluster appear to be different from the properties of the corresponding two-phase equilibrium system.

1018

1017

16 ) 10 -1 s -3

J (m 1015

1014

1013 2.5 5 10 20 40 80

Sc9 40 bar, Peeters 25 bar, Luijten 33 bar, Labetski 10 bar, Luijten

Fig. 5.2: n-Nonane nucleation rate as a function of supersaturation and pressure. Experi- mental data (markers) the quasi-unary nucleation theory (lines).

Temperature and pressure correction

It is not possible to determine nucleation rates exactly at the same temperature and pressure without any deviations. These deviations can be corrected for and the data obtained for a particular nucleation isotherm can be reduced to data corresponding to some ﬁxed temperature and pressure. Such correction for temperature deviations was done by Looijmans [5]. The nucleation rate is a function of temperature, pressure, and supersaturation, which means that the deviations in the supersaturation for small deviations of pressure and temperature at constant nucleation rate can be written as

∂S δT = δS (5.1) ∂T J and ∂S δT = δP. (5.2) ∂P J 94 5.1. N-NONANE NUCLEATION IN METHANE

To estimate the values of the partial derivatives, the rather general expression for nucleation rates can be employed

(σ/T )3 J = A exp C 2 , (5.3) − (ln S) ! where the kinetic pre-factor A and constant C are assumed not to depend on pressure, temperature and supersaturation. At constant J equation (5.3) gives the im- plicit dependence of supersaturation on pressure and temperature. The following related expressions are useful:

∂S 2 S ln S T ∂σ = 1 (5.4) ∂T 3 T σ ∂T − J P and ∂S 2 S ln S ∂σ = . (5.5) ∂P 3 σ ∂P J T To estimate the partial derivatives, it should be known how the surface tension changes with pressure and temperature. For a cluster, the dependence of surface tension or surface energy on temperature and pressure is not known at all, but it is reasonable to assume that the surface tension of a cluster depends on the temperature and pressure in a similar way as the surface tension of the corresponding two-phase system. Then, the equations give a way to estimate the correction of supersaturation and to reduce temperature and pressure to the chosen ones. In Fig. 5.3, the experimental data with corrected supersaturation is shown together with the original data. The correction decreases the scatter of the data, especially for low pressures.

1018 original data S corrected data

1017

) 10 -1 s -3

J (m 1015

1014

1013 2.5 5 10 20 40 80

Sc9

Fig. 5.3: n-Nonane nucleation data corrected for temperature and pressure deviations.

95 CHAPTER 5. RESULTS AND DISCUSSION

400

300

200 (ppm) c9 y

100

0 5 10 20 40 80 P (bar)

Fig. 5.4: Initial n-nonane molar fraction as a function of pressure at constant nucleation rate 16 −3 −1 10 m s and temperature 240 K.

Critical cluster composition

The nucleation theorem extended to the multicomponent system can be applied to the presented experimental data. According to the nucleation theorem (Sec. 3.3), in order to estimate the excess number of molecules in the critical cluster one should know the slope of the nucleation isotherms in the logarithmic J-S plot and the dependence of the initial molar fraction of a vapor component on pressure at given temperature and nucleation rate. All this information can be extracted from the available experimental data. To find the y0,c9 dependence on pressure, the following procedure has been employed. The initial molar fraction should be found 16 1 3 for some constant value of the nucleation rate, say at 10 s− m− , which is just in the middle of the data points. Then, the initial molar fraction of n-nonane is determined for a given pressure at the fixed nucleation rate. In such a way, the dependence of y0,c9 on P at constant J and T can be found. It appears, that values obtained in this way are linearly dependent on ln P as shown in Fig. 5.4. It is not yet clear, whether this correlation is a matter of physical principle. This linear dependence can be employed to find accurate values for(∂ ln y0,c9/∂ ln P )J,T at given J and P . Finally, for the partial derivatives found, the excess numbers of molecules in the critical cluster can be derived, as the solution of equations (3.73) and (3.77): ∂ ln J = ∆n∗ + 1 (5.6) ∂ ln S c9 c9 P,T and gas gas P v˜c9 P v˜c1 ∂ ln y ∆nc∗9 + ∆nc∗1 + 2 ln fe 0,c9 = kbT kbT . (5.7) ∂ ln P − ∆n + 1 J,T c∗9

96 5.1. N-NONANE NUCLEATION IN METHANE

50 ∆nc1 ∆n 45 c9 ∆ntotal 40

n 25 ∆

0 5 10 15 20 25 30 35 40 45 P (bar)

Fig. 5.5: Critical cluster composition of methane/n-nonane as a function of pressure at 240 K.

The results of the calculations for methane/n-nonane system are shown in Fig. 5.5. The ﬁrst conclusion is that the composition and size of the critical cluster do depend on pressure. With pressure increase the number of n-nonane molecules in the critical cluster is decreasing; the number of methane molecules, however, is increasing between 10 and 25 bar and then it drops when the pressure exceeds 25 bar. The total number of molecules in the critical cluster is also increasing but then drops for pressures above 25 bar. At the highest pressure, the critical cluster contains about 4 n-nonane and 24 methane molecules. The nucleation rate data unfortunately does not give the possibility to speculate about the structure of the critical cluster with such compositions.

For a methane/n-nonane system at the experimental conditions considered the excess numbers of molecules are close to the absolute numbers of n-nonane and methane molecules in the critical cluster. Because n-nonane is a nucleating component, the cluster contains an appreciable amount of n-nonane molecules, but in the gas phase the n-nonane molar fraction is of the order of hundreds ppm, obviously ∆n∗ n∗ . The number of methane molecules in the cluster is related to the c9 ≈ c9 excess number of methane molecules and with the number of n-nonane molecules as liq v˜c9 nc∗1 ∆nc∗1 + nc∗9 gas , (5.8) ≈ v˜c1 where it was supposed that the molar volume of critical cluster can be replaced by the partial molar volumes of methane and n-nonane for the bulk liquid phase at the same conditions, and the inequality v˜liq/v˜gas 1 was used. So, it is possible to c1 c1 ≪ calculate the absolute number of methane and n-nonane molecules in the critical cluster. In Tab. 5.1 the composition of the critical clusters is presented at different

97 CHAPTER 5. RESULTS AND DISCUSSION pressures. The estimate reveals that the composition of the critical clusters is also changing, the relative number of methane molecules is increasing with pressure. In the table the equilibrium composition for corresponding two-phase system is

eq P (bar) nc∗9/(nc∗9 + nc∗1) xc9 10.1 0.81 0.93 25.1 0.31 0.83 33.5 0.20 0.78 40.1 0.13 0.75

Tab. 5.1: Composition of critical cluster and equilibrium liquid phase. also listed. A similar tendency of decreasing of the n-nonane molar fraction with pressure is observed for liquid phase at equilibrium, but the effect is signiﬁcantly less than for the critical cluster composition. At a pressure of 10 bar the compositions of the critical clusters and the equilibrium liquid phase compositions are close to each other, and it can be expected that the thermodynamic properties of the cluster are close to the properties of the equilibrium mixture. This explains a good agreement between the quasi-unary theory and the experimental observations at low pressures. But this is certainly not the case for high pressures, where the large difference between the cluster composition and the equilibrium composition of two-phase system puts under question the use of any nucleation theory based on the bulk liquid phase properties.

Droplet growth data

Apart from nucleation rates, the droplet growth rates were measured. The data is listed in the corresponding table in Appendix C and is shown in Fig. 5.6. The droplet growth data are in the form of the droplet surface growth rates as a function of n-nonane molar fraction at a given temperature, and pressure. The droplet surface growth rate was obtained for each experiment by linear regression of the 2 rd(t) data, taking the time origin in the middle of the nucleation pulse. The droplet growth temperature and pressure were 246.7 0.7 K and 37.5 0.4 bar. ± ±

The droplet growth rate data allows to deduce the value of the diffusion coefficient and the equilibrium n-nonane molar fraction in the gas phase at droplet growth conditions. From the expression for the droplet growth rate for diffusion- controlled droplet growth (3.109), it follows that the slope of the linear relation be- 2 tween drd/dt and y0,c9 yields the diffusion coefficient for n-nonane in methane for known values of the equilibrium n-nonane molar fraction, molar densities of the gas and liquid phases. These values are evaluated from the RKS-EOS, and the in- 7 2 terpolation of the data gives 1.47 10− m /s for the diffusion coefficient D . The · c9,c1 7 2 Fuller [6] correlation for the diffusion coefficient predicts 1.46 10− m /s, which · agrees quite well with the value inferred from the data. It should be noted how-

98 5.2. N-NONANE NUCLEATION AND DROPLET GROWTH IN METHANE/PROPANE CARRIER GAS MIXTURES

12.0 model exp eq 10.0

8.0 /s) 2 m

µ 6.0 /dt ( 2 d dr 4.0

2.0

0.0 0 20 40 60 80 100 120

y0,c9 (ppm)

Fig. 5.6: Surface growth rate as a function of n-nonane initial molar fraction at 246.7 K and 37.5 bar. A closed squares indicates the equilibrium n-nonane molar fraction.

ever, that the Fuller expression is accurate within 10%. The analysis of the data for higher and lower pressures performed by Peeters et al. [7] shows that the measured and calculated diffusion coefﬁcient agree within the uncertainty of the Fuller correlation..

5.2 n-Nonane nucleation and droplet growth in methane/propane carrier gas mixtures

A series of experiments has been done on nucleation and droplet growth of n- nonane in methane/propane carrier gas mixtures. The nucleation rate data was obtained at temperatures and pressures around 240.3 K and 40.1 bar. The temperature in the nucleation pulse was obtained as described in Sec. 2.3 taking into account the presence of propane in the carrier gas. Droplet growth was studied at relatively higher values of temperature and pressure, approximately 247.8 K and 44.7 bar. The temperature and pressure in different experiments are varying in the following intervals 0.5 K and 0.3 bar. All obtained data are listed in the corre- ± ± sponding tables in Appendix C. The molar fraction of n-nonane has been varied in the range from 50 to 78 ppm. In most experiments the molar fraction of propane was 0.1, but some experiments were performed with mixtures with propane molar fraction of 0.03 and 0.05.

99 CHAPTER 5. RESULTS AND DISCUSSION n-nonane nucleation in methane/propane mixtures

Some of the obtained nucleation rate data is plotted in Fig. 5.7 (closed circles), as a function of n-nonane supersaturation Sc9. The data is for a methane/propane carrier gas mixture with a 0.01 molar fraction of propane. The n-nonane supersaturation has been calculated using the expression

eq Sc9 = y0,c9/yc9 (5.9) derived in Sec. A.3. The equilibrium molar fraction was obtained with the NIST database program, as a result of a ﬂash calculation at constant temperature and pressure for an initial mixture composition (y0,c1,y0,c3,y0,c9) equal to the experimental one. The equilibrium n-nonane molar fraction for experimental conditions is approximately 16 ppm, only slightly changing from one experimental point to another.

1018 C1/C3(0.01)/C9 C1/C9

1017 ) -1 s -3 J (m

1016

1015 1 2.5 5 10 15

Sc9

Fig. 5.7: n-Nonane nucleation rate in methane/propane(0.01) and in methane as a function of supersaturation at 240 K and 40 bar.

In Fig. 5.7, n-nonane nucleation rates in methane and in methane/propane(0.01) carrier gas mixtures are compared. The n-nonane nucleation rates observed in methane carrier gas are systematically lower than the nucleation rate observed in a methane/propane(0.01) mixture. Also it should be noted that for the same supersaturations of n-nonane in both mixtures, the actual molar fraction of n-nonane is different. In the methane/propane(0.01) mixture it is lower. This is partly explained by the fact that the equilibrium molar fraction of n-nonane in the presence of propane is lower, and according to (5.9) it gives a higher supersaturation when propane is added to a carrier gas mixture. It was shown earlier in Sec. 3.3, that the slope of the nucleation isotherm in a ln J-ln S plot gives the excess number of vapor molecules in the critical cluster. In this particular case, it equals the number of

100 5.2. N-NONANE NUCLEATION AND DROPLET GROWTH IN METHANE/PROPANE CARRIER GAS MIXTURES n-nonane molecules in the critical cluster because the molar fraction of n-nonane in the gas phase is very low, and the molar density of the gas phase is much lower than the molar density of the cluster. Therefore, it is a good approximation to say that the slope of the nucleation isotherm is equal to the number of n-nonane molecules in the critical cluster. By estimating the slope of the nucleation isotherm in Fig. 5.7, the number of n-nonane molecules in the critical cluster is estimated to be approximately 4.

A few experiments have been performed with higher molar fractions of propane in the carrier gas mixtures (0.03 and 0.05). Unfortunately, it was not possible to evaluate the nucleation rate of n-nonane for those mixtures. These experiments are listed in the tables in Appendix C. The scattering light intensity signals for methane/propane(0.03) and methane/propane(0.05) do not even show a ﬁrst maximum, which means that the evaluation of the droplet size is impossible with the method described in Sec. 2.4. What can be done to derive nucleation rates, is to suppose that the droplet growth law is known and independent of the droplet concentration. The transmitted intensity signal depends on the droplet number density, and so a comparison between the observed transmitted signal and the result of droplet growth calculation will give the number density of droplets. The transmitted light signal observed in the experiments is shown in Fig. 5.8. The accuracy of such method is rather low, but it gives qualitative estimates of the droplet growth rates. Such analysis was performed for transmitted intensity signals available for methane/propane(0.03) and methane/propane(0.05) mixtures. The analysis reveals that for the same supersaturations, the nucleation rates is increasing with an increase of the molar fraction of propane in the carrier gas mixture.

1 C1/C3(0.03)/C9 C1/C3(0.01)/C9 0.9

0.8

0.7

0.6 0 /I 0.5 trans I 0.4

0.3

0.2

0.1

0 0 5 10 15 20 25 30 35 t (ms)

Fig. 5.8: Transmitted intensity signals observed in different methane/propane/n-nonane mixtures at the same n-nonane supersaturation at 240 K and 40 bar.

The n-nonane nucleation rate in a methane/propane(0.01) carrier gas mixture

101 CHAPTER 5. RESULTS AND DISCUSSION is calculated using the quasi-unary nucleation rate expression (3.35) derived in Sec. 3.2. To perform the calculation, the energy of cluster formation in the two- phase equilibrium has to be known. An expression similar to (3.61) was used, containing two terms: surface energy and energy of mixing. The cluster surface tension is taken as the surface tension of a gas-liquid interface at phase equilibrium. The equilibrium surface tension for ﬂat interfaces is calculated with the Macleod- Sugden correlation, using the equilibrium thermodynamic properties from the NIST database. The obtained result is shown together with the experimental data in Fig. 5.9. The quasi-unary theory does not predict neither nucleation rate nor the slope of the nucleation isotherm. The energy of cluster formation is apparently different from our estimates.

1018 C1/C3(0.01)/C9 quasi-unary NT

1017 ) -1 s -3 J (m

1016

1015 1 2.5 5 10 15

Sc9

Fig. 5.9: n-Nonane nucleation rate data in methane/propane(0.01) and prediction of quasi- unary nucleation theory at 240 K and 40 bar.

n-Nonane droplet growth in methane/propane carrier gas

The droplet growth data are obtained at 247.8 K and 44.7 bar with some variations from one data point to another. For a particular experiment, the droplet growth pressure and temperature are taken as averages of pressure and temperature after the nucleation pulse. The data is shown in Fig. 5.10. The results are plotted as 2 surface growth rate drd/dt as a function of initial molar fraction of n-nonane y0,c9 in the gas phase. In these coordinates, the droplet growth data will lie on a straight line if the droplet growth is diffusion-controlled which is expected at the given experimental conditions (see Sec 3.4):

2 gas drd 2ρ Dc9 eq = eq liq (y0,c9 yc9 ) . (5.10) dt xc9ρ −

102 5.2. N-NONANE NUCLEATION AND DROPLET GROWTH IN METHANE/PROPANE CARRIER GAS MIXTURES

According to the diffusion controlled droplet growth law (5.10), the slope of the 2 droplet growth data in (drd/dt, y0,c9) coordinates is proportional to the diffusion coefficient of n-nonane in the carrier gas. The intersection of surface droplet growth 2 data with the ordinate axes at drd/dt=0 gives the equilibrium molar fraction of n- nonane at droplet growth conditions. By fitting the growth data with a straight line, the diffusion coefficient and phase equilibrium molar fraction are obtained: exp 7 2 exp D = 12.3 10− m /s and y = 33.0 ppm . The obtained value of the diffu- c9 × 0,c9 F 7 2 sion coefficient differs from the value D = 12.1 10− m /s estimated with the c9 × Fuller correlation. The difference is well within the relative uncertainty of the correlation which is about 0.15.

12.0 linear fit C1/C3(0.01)/C9 C1/C9 10.0

8.0 /s) 2

m 6.0 µ /dt ( 2 d

dr 4.0

2.0

0.0

0 20 40 60 80 100 120

yc9 (ppm)

Fig. 5.10: n-Nonane surface growth rate in methane and methane/propane(0.01) at 247.8 K and 44.7 bar. Open diamond marks equilibrium n-nonane molar fraction.

It is possible to calculate the equilibrium n-nonane molar fraction at droplet growth NIST conditions with an equation of state. This point, y0,c9 = 35.0 ppm, evaluated from NIST database, is denoted as an open diamond in Fig. 5.10. The relative difference between the measured and evaluated from data equilibrium n-nonane molar fractions and the values from the EOS-evaluation is less than 0.1. This is within the uncertainty of the equilibrium calculations caused by variations of temperature and pressure from one experiment to another.

In Fig. 5.10 the droplet growth data for the methane/n-nonane mixture is shown as open circles. The data was obtained by P. Peeters et al. [2] at about the same temperature and pressure as for n-nonane growth in the methane/propane(0.01) mixture. The presence of a 0.01 molar fraction of propane in the methane carrier gas does not change the diffusion coefﬁcient and the equilibrium molar fraction of n-nonane appreciably. Therefore, the droplet growth curves should match, and this is exactly what is observed. An important conclusion can be derived from the consistency of both data-sets: that the mixture preparation procedure used in both

103 CHAPTER 5. RESULTS AND DISCUSSION experiments is robust and gives repeatable initial mixture compositions.

5.3 n-Nonane nucleation and droplet growth in methane/carbon dioxide mixtures

We have seen that adding propane to the methane carrier gas has already a sig- nificant influence on nucleation of n-nonane even for a relatively small propane molar fraction of 0.01. Because of its abundant presence in natural gas the effect of carbon dioxide on n-nonane nucleation in methane has been investigated. We have restricted our study to carbon dioxide concentrations for which it remains subsaturated. First, experiments were performed with methane/carbon dioxide carrier gas mixtures with a 0.01 molar fraction of carbon dioxide at three different conditions: 236 K and 10 bar, 240 K and 25 bar, 240 K and 40 bar. Second, the nucleation rates of n-nonane in methane/carbon dioxide mixtures were determined at 235 K and 10 bar for initial molar fractions of carbon dioxide varying up to 0.3. Two series of such experiments have been performed: 1) the molar fraction of carbon dioxide is fixed, yco2 = 0.25, and the n-nonane fraction is varied; 2) the molar fraction of n-nonane is constant, yc9 = 230 ppm, and the carbon dioxide fraction is varied. All the data are listed in the corresponding tables in Appendix C.

n-Nonane nucleation rate in methane/carbon dioxide

The new data on n-nonane nucleation in a methane/carbon dioxide(0.01) mixture are shown as nucleation isotherms in Fig. 5.11. The actual temperature and pressure for a particular data point can differ from the average values depicted in the

figure within 0.9 K and 0.3 bar intervals. The supersaturation of n-nonane Sc9 ± ± eq is calculated with (5.9) and the equilibrium molar fraction of n-nonane yc9 is a result of a flash calculation with the NIST database program. The first conclusion from Fig. 5.11 is that with pressure increase the nucleation isotherms shift to lower supersaturations, the same trend as observed for methane/n-nonane (see Fig. 5.1). So, by analogy with the methane/n-nonane nucleation experiments it can be expected that the concentration of gas molecules, methane or/and carbon dioxide, in the critical cluster is increasing with pressure. Also, the slopes of the nucleation isotherms are decreasing with pressure increase. According to the nucleation theorem, this indicates that the number of n-nonane molecules in the critical cluster is decreasing. Because of the large scatter of data at 40 bar it is difficult to define accurately the slope of the isotherm and to obtain the exact number of n-nonane molecules. This large scatter of data is caused by the scatter in the nucleation temperatures. In the experiments at lower pressures, P = 25 and 10 bar, the tem-

104 5.3. N-NONANE NUCLEATION AND DROPLET GROWTH IN METHANE/CARBON DIOXIDE MIXTURES perature scatter is lower.

1018

1017

) 10 -1 s -3

J (m 1015

1014 236 K and 10 bar 240 K and 25 bar 240 K and 40 bar 1013 1 2.5 5 10 20 40 80

Sc9

Fig. 5.11: n-Nonane nucleation rate in methane/carbon dioxide(0.01) as a function of n- nonane supersaturation.

In Fig. 5.12 the n-nonane nucleation data for methane/carbon dioxide(0.01) are plotted together with the methane/n-nonane data obtained by Luijten [1] and by P. Peeters et al. [2]. The nucleation isotherms for both datasets at 10 and 25 bar almost coincide. At 40 bar the nucleation rates in methane are systematically higher than in methane/carbon dioxide. Also, it should be noted that the addition of a 0.01 molar fraction of carbon dioxide to the methane carrier gas almost does not change the eq equilibrium n-nonane molar fraction yc9 in the gas phase. For example, at 240 K eq and 40 bar the equilibrium molar fraction yc9 is 18.0 ppm in methane/carbon diox- eq ide(0.01) and yc9 = 17.9 ppm in methane. As a result, the data shown in Fig. 5.12 are obtained at the same initial molar fractions of n-nonane. Furthermore, phase equilibrium calculations show that a relatively small amount of carbon dioxide will dissolve in liquid n-nonane, say for 240 K and 40 bar the molar fraction of carbon dioxide in the liquid phase is approximately 0.011. The same trend can be expected for the critical cluster composition. Certainly, even a rather small amount of carbon dioxide in the cluster can inﬂuence the energy of cluster formation and so the nucleation rate. But, the observed nucleation rates suggest that this is not the case at least for low carbon dioxide concentration.

Next, the effect of an increase of carbon dioxide concentration on n-nonane nucleation has been studied. Two series of experiments have been done for n-nonane nucleation in methane/carbon dioxide with relatively high, up to 0.3, molar fractions of carbon dioxide. The results of these experiments are shown in ﬁgure 5.13. At the same supersaturation of n-nonane, the nucleation rate of n-nonane (close circles) in methane/carbon dioxide(0.25) is approximately one order of magnitude higher than the n-nonane nucleation rate (open squares) in methane. Therefore,

105 CHAPTER 5. RESULTS AND DISCUSSION

1018

1017

) 10 -1 s -3

J (m 1015

1014

C1/CO2(0.01)/C9 C1/C9 1013 1 2.5 5 10 20 40 80

Sc9

Fig. 5.12: n-Nonane nucleation rate in methane and methane/carbon dioxide(0.01) as a function of supersaturation.

the presence of carbon dioxide stimulates nucleation of n-nonane. If the molar fraction of n-nonane is kept constant and we change the composition of the carrier gas by increasing the molar fraction of carbon dioxide, the nucleation rate of n-nonane will increase (close triangles). A higher molar fraction of carbon dioxide in the carrier gas leads to a higher supersaturation of n-nonane Sc9. This is because the addition of carbon dioxide decreases the n-nonane equilibrium molar fraction eq eq yc9 and so increases supersaturation according to Sc9 = y0,c9/yc9 . The increase of the nucleation rate with supersaturation is a trivial fact, but what is new and not trivial, is that the change of the n-nonane supersaturation by adding carbon dioxide affects the nucleation rate more strongly than the change of supersaturation by adding n-nonane. Evidently, an addition of carbon dioxide to the carrier gas changes not only the equilibrium state, but it also changes the energy of critical cluster formation.

With the quasi-unary nucleation theory, the nucleation rates are evaluated for the cases discussed. For the evaluation the general quasi-unary expression (3.35) for the nucleation rate has been used. To estimate the energy of cluster formation the expression with mixing and surface terms was taken. As a surface tension of a cluster, the ﬂat interface equilibrium surface tension was used with the Macleod-Sugden correlation. We were not able to apply here the semi-empirical Parachor method, since accurate surface tension data for ternary methane/carbon dioxide/n-nonane mixture is not available. The consequence is that the absolute values of the nucleation rates are much too low. Therefore, the theoretical nucleation rates should be considered as indicative for their dependence on supersaturation and carbon dioxide concentration.

106 5.3. N-NONANE NUCLEATION AND DROPLET GROWTH IN METHANE/CARBON DIOXIDE MIXTURES

1018

1017

) 10 -1 s -3

J (m 1015

1014

yco2 = 0.00 yco2 = 0.25 yc9 = 230 ppm 1013 10 15 25 45 85

Sc9

Fig. 5.13: n-Nonane nucleation rate in methane/carbon dioxide mixtures and in methane at 235 K and 10 bar.

The results are presented in Fig. 5.14. The values of the theoretical nucleation rates predicted by the quasi-unary theory are several orders of magnitude lower than the experimental values (see Fig. 5.12). The qualitative behavior is predicted correctly: the addition of carbon dioxide to the carrier gas mixture facilitates n- nonane nucleation. The slopes of the experimental and theoretical nucleation rates are approximately the same. This means that the numbers of n-nonane molecules in the critical cluster are well predicted. The relative increase of the n-nonane nucleation rate as a result of adding 0.25 molar fraction of carbon dioxide to methane is experimentally in between 10-20. The theory predicts a factor 100 increase which is signiﬁcantly different. The effect of adding carbon dioxide for a ﬁxed n-nonane concentration is qualitatively well predicted by theory.

Droplet growth of n-nonane in methane/carbon dioxide

In the previous section it was shown that for n-nonane droplet growth the presence of a 0.01 molar fraction of propane in the methane carrier gas does not appreciably affect droplet growth. The same observation holds for a methane/carbon dioxide carrier gas with a low (0.01) molar fraction of carbon dioxide. By adding more and more carbon dioxide to methane, it is possible to ﬁnd conditions for which the droplet growth will be affected.

At 242 K and 11 bar, the droplet growth rate is diffusion controlled, so it satis- ﬁes: 2 gas drd 2ρ Dc9 eq = eq liq (y0,c9 yc9 ) . (5.11) dt xc9ρ −

107 CHAPTER 5. RESULTS AND DISCUSSION

1018 yco2 = 0 yco2 = 0.25 yc9 = 230 ppm 1016

1014 ) -1 s -3 1012 J (m

1010

108

10 15 25 45 85

Sc9

Fig. 5.14: n-Nonane nucleation rate as a function of supersaturation: experiment (markers) and theory (lines), at 235 K and 10 bar. Circles and squares: ﬁxed carbon dioxide concentration. Triangles: ﬁxed n-nonane concentration. For details, see Fig. 5.13.

25.0 yco2 = 0 yco2 = 0.25

20.0 /s) 2 m

µ 15.0 /dt ( 2 d dr

10.0

5.0 100 150 200 250 300

yc9 (ppm)

Fig. 5.15: n-Nonane surface growth rate in methane and in methane/carbon dioxide(0.25) at 242 K and 11 bar.

108 REFERENCES

The addition of carbon dioxide to the methane carrier gas will change the phase equilibrium properties, namely, gas phase and liquid phase molar densities ρgas liq eq eq and ρ , as well as phase compositions xc9 and yc9 . The molecular diffusion coefficient of n-nonane in the carrier gas mixture can be estimated with Blank’s law [6]: 1 y y − D = co2 + c1 , (5.12) c9 D D co2,c9 c1,c9 where Dco2,c9 and Dc1,c9 are the molecular diffusion coefficients of n-nonane in pure carbon dioxide and methane, respectively. The molar diffusion coefficients of n-nonane in methane and carbon dioxide are different, the Fuller correlation gives 7 2 7 2 the following values: D = 2.49 10− m /s and D = 4.13 10− m /s. co2,c9 × c1,c9 × Therefore, with addition of carbon dioxide the diffusion of n-nonane molecules slows down approaching diffusion in pure carbon dioxide. The comparison of experimental data with the diffusion-controlled droplet growth law is shown in Fig. 5.15. The observed surface growth rate of n-nonane in methane/carbon dioxide(0.25) is systematically lower than in pure methane, as it is predicted with the diffusion-controlled droplet growth law.

References

[1] C.C.M. Luijten, P. Peeters, and M.E.H. van Dongen, J. Chem. Phys. 111, 8535 (1999).

[2] P. Peeters, J. Hrubý, and M.E.H. van Dongen, J. Phys. Chem. B 105, 11763 (2001).

[3] C.C.M. Luijten and M.E.H. van Dongen, J. Chem. Phys. 111, 8524 (1999).

[4] J.R. Deam and R.N. Maddox, J. Chem. Eng. Data 15, 216 (1970).

[5] K.N.H. Looijmans, Homogeneous nucleation and droplet growth in the coexistence region of n-alkane/methane mixtures at high pressures, PhD thesis, Eindhoven Uni- versity of Technology, 1995.

[6] R.C. Reid, J.M. Prausnitz, and B.E. Poling, The Properties of Gases and Liquids, McGraw–Hill Book Company, New York, 1987.

[7] Peeters P, Pieterse G, Hruby J, and van Dongen MEH, Phys. Fluids 16, 2567 (2004).

109 CHAPTER 5. RESULTS AND DISCUSSION

110 Chapter 6

Conclusions and recommendations

Nucleation and droplet growth processes play an important role in the separation of contaminants from natural gas. In this thesis several model systems of natural gas have been considered: methane/n-nonane, methane/propane/n-nonane, and methane/carbon dioxide/n-nonane. Nucleation has been studied with an expansion wave tube. In this tube, the nucleation pulse principle is applied to split the nucleation and droplet growth processes in time. The gas-vapor mixture under study is adiabatically expanded, which brings the mixture into a supersaturated state, such that nucleation occurs. After a short period of time, the vapor supersaturation is slightly decreased by re-compression. Nucleation is effectively stopped, but droplet growth remains possible, because the vapor is still supersaturated. The droplets are detected by optical means using a combination of Mie scattering techniques. Droplet number densities, nucleation rates, and the thermodynamic state variables have been measured. The gas-vapor mixtures have been prepared in the mixture preparation part of the setup with a relatively high accuracy. The setup has been modiﬁed to allow the preparation of ternary mixtures. By varying the composition of the gas-vapor mixture, the nucleation processes have been studied at different conditions, and information about effects of non-condensable gases on nucleation have been deduced.

In interpreting nucleation rate data, the nucleation theorem formulated by Ox- toby and Kashchiev is of great value. We have re-formulated this theorem such that the composition of the critical cluster can be deduced from experimental nucleation rate data for binary mixtures and known thermodynamic properties. Fur- ther, it was shown that for a diluted vapor in a carrier gas, the kinetics of nucleation reduces to a quasi-unary form.

111 CHAPTER 6. CONCLUSIONS AND RECOMMENDATIONS

We have continued the systematic study of n-nonane nucleation in methane. Ad- ditional experimental data has been obtained at 240 K and 33 bar. The experimental data conﬁrms the strong inﬂuence of the methane carrier gas on nucleation: an increase of pressure strongly facilitates nucleation, and alters the composition of the critical cluster. The numbers of methane and n-nonane molecules in the critical cluster have been evaluated by applying the nucleation theorem. It was found that at 40 bar the critical cluster contains approximately 4 n-nonane and 23 methane molecules. With pressure decrease, the number of n-nonane molecules in the critical cluster is increasing and the methane fraction in the cluster decreases, approaching the equilibrium methane molar fraction in bulk liquid. Evidently, the capillarity approximation, widely used in nucleation theories, cannot be applied to methane/n-nonane at 40 bar.

A series of experiments has been performed in methane/propane carrier gas mixtures with different concentrations of propane at 240 K and 40 bar. The data was obtained at conditions at which they can be compared with available data for methane/n-nonane mixtures. It is found that a 0.01 molar fraction of propane in methane is enough to increase the n-nonane nucleation rate by 5-7 times in comparison with the n-nonane nucleation rate in pure methane. Unfortunately, it was not possible to obtain accurate nucleation data for higher concentrations (0.03 and 0.05) of propane, but a qualitative analysis of the transmitted light intensity signals, based on known droplet growth, shows a further increase of the nucleation rate with the molar fraction of propane.

An extensive study of n-nonane nucleation in methane/carbon dioxide has been performed. Nucleation rate data was obtained in methane/carbon dioxide carrier gas mixtures with a 0.01 molar fraction of carbon dioxide at three different conditions: 236 K and 10 bar, 240 K and 25 bar, 240 K and 40 bar. By comparison of the obtained data with the available data for the methane/n-nonane mixture, it is found that for such low concentrations of carbon dioxide, the influence on nucleation rate is observable though rather weak. Furthermore, the nucleation rates of n-nonane in methane/carbon dioxide mixtures were determined at 235 K and 10 bar for initial molar fractions of carbon dioxide up to 0.3. Two series of such experiments have been performed: 1) the molar fraction of carbon dioxide is fixed, yco2 = 0.25, and the n-nonane fraction is varied; 2) the molar fraction of n-nonane is constant, yc9 = 230 ppm, and the carbon dioxide fraction is varied. A comparison with the methane/n-nonane data shows that carbon dioxide does influence nucleation when its molar concentration in the mixture exceeds 0.25. At the same supersaturation of n-nonane, the nucleation rate in methane/carbon dioxide(0.25) is approximately one order of magnitude higher than the n-nonane nucleation rate in methane.

4 For a dilute vapor (molar fraction 10− ) in the presence of a carrier gas mixture the quasi-unary nucleation theory applies. In the quasi-unary expression for the nu-

112 cleation rate, the unknown parameter is the equilibrium cluster size-distribution deﬁned by the energy of cluster formation. The energy of cluster formation was deduced on the basis of the capillarity approximation. The comparison with experimental rate data shows that the quasi-unary theory yields nucleation rates comparable with experimental ones at rather lower pressures ( 10 bar). But at high pres- ≈ sures ( 40 bar) the theory neither predicts nucleation rates nor the composition of ≈ critical clusters correctly. This is consistent with the experimental observation that the critical cluster composition at high pressure substantially differs from the equilibrium composition of the corresponding equilibrium liquid mixture.

The gradient theory of the phase interface offers an approach to determine density distributions of mixture components in the interface and the energy of surface formation if the bulk thermodynamic properties are known for both phases at equilibrium. This theory can also be applied to a critical cluster which is in a metastable equilibrium. It yields the size, the structure and the energy of formation for the critical cluster. The theory is not yet fully established and needs further development, but seems very promising in particular for multi-component mixtures.

The present work has shown that the physics of nucleation in complex natural gas-like mixtures is not well understood so far. Strong interaction exists between "vapor" molecules and molecules of the carrier gas, such that the critical cluster may have properties far from those of any known equilibrium state. In particular at pressures above 25 bar, methane tends to play a very active role in hydrocarbon nucleation. There is a strong need both for further accurate and reliable experiments at pressures exceeding 25 bar and for theoretical modeling of the properties of small clusters in multicomponent mixtures.

113 CHAPTER 6. CONCLUSIONS AND RECOMMENDATIONS

114 Appendix A

Phase equilibrium and surface tension in multicomponent mixtures

A.1 Phase equilibrium in two-phase system

The thermodynamic equilibrium in a two-phase system can be described using the chemical potential formalism. The chemical potential is a thermodynamic state potential which is prescribed for each component in each phase and depends on temperature, pressure and composition. Then, phase equilibrium is deﬁned as a state in which the chemical potentials of the components are equal in both phases: α β µi (T,P,y1,y2, ..., yn) = µi (T,P,x1,x2, ..., xn) , (A.1)

α β where µi and µi are the chemical potentials of the i-th component in the gas phase and in the liquid phase, respectively. In equilibrium the equality holds for all n components.

According to Gibbs’ phase rule, the number of independent intensive variables for a two phase system is the same as the number of components in the mixture. Therefore, by solving the system of equations (A.1) the equilibrium pressure and the equilibrium composition of the liquid phase can be found if the temperature, and the equilibrium composition of the gas phase are speciﬁed.

The chemical potential can be calculated if the equation of state is known [1]. In this study we used the RKS and PR equations of state with quadratic mixing APPENDIX A. PHASE EQUILIBRIUM AND SURFACE TENSION IN MULTICOMPONENT MIXTURES rules to describe liquid and gas phases. In some cases, available phase equilibrium software PE2000 [2] and SUPERTRAPP [3] are used to study phase equilibrium in multicomponent mixtures.

A.2 Equations of state

The Redlich-Kwong-Soave (RKS) equation of state has the form:

R T b P = a , (A.2) V a − V (V + a) − where 0.08664R T a = a c (A.3) Pc and 2 0.42748R2T T b = a c 1 + f(ω) 1 (A.4) P T c − r c !! with f(ω) = 0.48 + 1.574 ω 0.176 ω2. (A.5) −

The Peng-Robinson (PR) equation of state is a cubic equation similar to RKS:

R T b P = a , (A.6) V a − V 2 + 2 aV a2 − − where 0.07780R T a = a c (A.7) Pc and 2 0.45724R2T T b = a c 1 + f(ω) 1 (A.8) P T c − r c !! with f(ω) = 0.37464 + 1.54226 ω 0.26992 ω2. (A.9) −

In the above equations, Ra is the universal gas constant, V is the molar volume, ω is Pitzer’s acentric factor, and the subscript c refers to the critical point. For a mixture, a and b are evaluated from the pure component values using the quadratic mixing rules:

a = ai yi (A.10) i X and b = y y (b b )1/2(1 k ) . (A.11) i j i j − i,j i j X X 116 A.3. SUPERSATURATION OF VAPOR DILUTED IN THE CARRIER GAS

In the above expressions yi denotes the molar fraction of the i-th component; ki,j is an interaction parameter, its value determined by a ﬁt to experimental phase equilibrium data in binary mixtures. For the large amount of binary mixtures, interaction parameter ki,j is found and listed in [4].

A.3 Supersaturation of vapor diluted in the carrier gas

The chemical potential µv of vapor in the gaseous phase can be written as [5]:

φvyvP µv = µv⊖ + kbT ln , (A.12) v⊖ F where superscript denotes a reference state, y is the vapor molar fraction in the ⊖ v mixure, is the vapor fugacity and φ is the vapor fugacity coefﬁcient. The super- Fv v saturation S is deﬁned through the difference of the chemical potentials between 0 eq the initial supersaturated state µv and the equilibrium µv ,

µ0(T,P,y ) µeq(T,P,y ) S = exp v v − v v . (A.13) k T b Because the vapor is diluted, its chemical potential does not include a dependence on the molar fractions of the carrier gas component. Combining these two expressions, the supersaturation can be written as:

0 0 φvyv S = eq eq . (A.14) φv yv

The fugacity coefficient φ characterizes intermolecular interactions. Since the vapor is diluted, vapor molecules will be interacting almost exclusively with carrier gas molecules. The fugacity coefficient in the supersaturated and in the equilibrium state will be approximately the same, therefore (A.14) can be simplified to 0 yv S = eq . (A.15) yv The initial molar fraction is known and the equilibrium molar fraction is found by applying an appropriate equation of state.

A.4 Surface tension in mixtures

The theory of nucleation predicts a strong dependence of the nucleation rate on the surface tension of the critical cluster, J exp ασ3 . As an approximation ∼ − of the critical cluster surface tension, the surface tension of the liquid phase at

117 APPENDIX A. PHASE EQUILIBRIUM AND SURFACE TENSION IN MULTICOMPONENT MIXTURES the same condition and in equilibrium state can be applied. The Macleod-Sugden correlation [5] can be used to calculate the surface tension of the liquid mixture in the equilibrium state σ1/4 = [P ](ρ x ρ y ) , (A.16) i L i − G i i X where σ is the surface tension of mixture, mN/m; [Pi] is the parachor of the i-th component; xi, yi are mole fractions of i-th component in liquid and gas phases; 3 ρL and ρG are the liquid and gas phase densities, mol/cm .

The parachors for the studied substances (methane, propane, carbond dioxide, and n-nonane) can be obtained from structural contribution tables [6] or from available surface tension experimental data, as it is done in [7] for the methane/n- nonane system.

References

[1] J.M. Prausnitz, R.N. Lichtenthaler, and E. Gomes de Azevedo. Molecular Ther- modynamics of Fluid-phase Equilibria. Prentice-Hall, New Jersey, 2nd edition, 1986.

[2] O. Pfohl, S. Petkov, and G. Brunner. Usage of PE - A program to Calculate Phase Equilibria. Herbert Utz Verslag, München, 1st edition, 1998. ISBN 3-89675-410- 6; http://www.tu- harburg.de/vt2/pe2000/HomePage.html.

[3] M. L. Huber. NIST Thermophysical Properties of Hydrocarbon Mixtures Data- base (SUPERTRAPP). 2003. http://www.nist.gov/srd/webguide/4-3.1/2- 04_3.1.htm.

[4] H. Knapp, R. Döring, L. Oellrich, U. Plöcker, and J.M. Prausnitz. Vapor– Liquid Equilibria for Mixtures of Low Boiling Substances. Deutsche Gesellschaft für Chemisches Apparatewesen, Frankfurt am Main, 1982.

[5] R.C. Reid, J.M. Prausnitz, and B.E. Poling. The Properties of Gases and Liquids. McGraw–Hill Book Company, New York, 1987.

[6] Osborne R. Quayle. The parachors of organic compounds: An interpretation and catalogue. Chem. Rev., 53(3):439–589, 1953.

[7] J.R. Deam and R.N. Maddox. Interfacial tension in hydrocarbon systems. J. Chem. Eng. Data, 15(2):216–222, 1970.

118 Appendix B

Simultaneous nucleation and droplet growth in the expansion wave tube

B.1 Characteristic depletion time

In the expansion wave tube, the nucleation and droplet growth processes are de- coupled in time. The decoupling is achieved with the nucleation pulse method. The method is based on the fact that the nucleation rate is a very steep function of the supersaturation; because of the steepness, most of the nuclei are formed in the pulse, after the pulse the nucleation is quenched and formed nuclei start to grow.

The growth of nuclei to macroscopic droplets causes vapor depletion. At certain conditions the characteristic time of vapor depletion is comparable with the duration of the nucleation pulse, the vapor depletion changes the supersaturation, and so it inﬂuences the nucleation rate: the nucleation pulse method fails. To estimate the characteristic depletion time, we use the expressions from [1]:

3 r t = G( ), (B.1) 4πD n0 rmax rmax

1+2x 2 π arctan( √ ) 1 + x + x G(x) = 3 + ln (B.2) 6√3 − √3 (1 x)2 − APPENDIX B. SIMULTANEOUS NUCLEATION AND DROPLET GROWTH IN THE EXPANSION WAVE TUBE

1000 10 Droplet radius, r ( radius, Droplet

100 1 m) Characteristicdepletion (ms)time, t

10 0.1

1 2 3 4 5

Fig. B.1: The characteristic depletion time t and droplet radius r at time t as a function of supersaturation. The supersaturation SC9 is at 247 K and 44 bar, the equilibrium n-nonane molar fraction yeq,c9 at this condition is 22.13 ppm.

1/3 3 ρG rmax = yeq (S(0) 1) (B.3) 4πn0 ρL − h i where D is the diffusion coefﬁcient, n0 is the number density of nuclei, rmax is the maximum droplet size, r is the droplet size, and t is the time spent by droplets to reach size r. As a characteristic time for depletion we took the time needed to form the droplets which accumulate 1/10 of all vapor available at given temperature 3 and pressure conditions; this means that r/rmax = 1/10. p The characteristic depletion times are estimated for methane/n-nonane mixtures and methane/water mixtures. We suppose that the droplet growth occurs at a temperature of 247 K and a pressure of 44 bar, the nucleation takes place in the pressure pulse at 40 bar and 240 K, the pulse duration is 0.45 ms (typical for our experiments at these conditions). The nucleation rate is calculated with an interpolation formula based on the data by P. Peeters [2].

The results of the calculations are shown in figure B.1 for the methane/n-nonane mixture and in figure B.2 for the methane/water mixture. The typical measurement time in the experiments is 35 ms. As one can see from figure B.1 the characteristic depletion time of n-nonane vapor is greater for the whole range of supersaturations achievable for n-nonane. We can conclude that the vapor depletion is not important for methane/n-nonane nucleation experiments.

The situation is different for water growing in methane, see ﬁgure B.2. The depletion of the vapor starts to inﬂuence droplet growth already at supersaturations

120 B.2. SIMULTANEOUS NUCLEATION AND DROPLET GROWTH MODEL

100 10 Droplet radius, r ( radius, Droplet

-1

10 10

-2

1 10 m) Characteristicdepletion (ms)time, t

-3

0.1 10

2.5 5.0 7.5

H2O

Fig. B.2: The characteristic depletion time t and droplet radius r at time t as a function of supersaturation . The supersaturation SH2O is at 247 K and 44 bar, the equilibrium water molar fraction yeq,h2o at this condition is 24.18 ppm.

higher than 4. Moreover, at supersaturations higher than 6.5 the characteristic depletion time is comparable with the pulse duration, which means that depletion of vapor inﬂuences the nucleation process. Therefore, for our experimental setup the pulse technique is not applicable at these conditions.

B.2 Simultaneous nucleation and droplet growth model

To study simultaneous nucleation and droplet growth, a simple model has been developed. In this model a series of assumptions is made:

1. the nucleation rate only depends on the supersaturation of vapor,

2. the droplet growth is diffusion controlled,

3. the droplets with sizes smaller than the size of the critical cluster are unstable and have to disintegrate.

121 APPENDIX B. SIMULTANEOUS NUCLEATION AND DROPLET GROWTH IN THE EXPANSION WAVE TUBE

These assumptions result in the following set of equations:

1 D ρG r˙ = (y yeq), (B.4) r xeq ρL − x ρ 4π ∞ y = y(0) eq L f(r, t)r3dr, (B.5) − ρ 3 G Z0 ∂f ∂ = (f r˙), (B.6) ∂t −∂r f r˙ = J, (B.7) |r=rcrit f = 0. (B.8) |r

Equation (B.4) deﬁnes the diffusion-controlled droplet growth rate; balance equation (B.5) ensures that the amount of vapor in the system remaining constant; equation (B.6) describes how the droplet size distribution function f(r, t) is changing with time; boundary condition (B.7) indicates that new droplets are appearing in the system with nucleation rate J and sizes equal to critical radius rcrit; and the last expression (B.8) states that the droplets with radius smaller than critical radius rcrit do not exist.

The system of equations (B.4–B.8) can be solved numerically, and the droplet size distribution can be obtained at any given time. To compare the model prediction with experimental data we have to calculate the transmitted intensity signal which is produced by the predicted droplet distribution for the geometry of the experimental setup. The transmitted intensity can be obtained for a given droplet size distribution f(r, t) from the following expression:

∞ I = I exp( L π f(r, t) Q r2dr). (B.9) trans 0 − ext Z0 Now, the predicted transmitted signal can be compared with the measured transmitted signal.

B.3 Comparison of model predictions with experimental data

Methane/n-nonane mixture

Tab. B.1: Experimental conditions exp Pnucl Tnucl yc9 yh2o Pgrowth Tgrowth (bar) (K) (ppm) (ppm) (bar) (K) 01nov05 001 40.0 240.0 99.6 0.0 44.9 247.3 03nov05 001 40.5 240.5 0.0 201.0 45.0 247.4

122 B.3. COMPARISON OF MODEL PREDICTIONS WITH EXPERIMENTAL DATA

Using the available pressure and temperature profiles, the droplet size distributions were calculated for the methane/n-nonane mixture, 01nov05_001 experiment. More details about experimental conditions can be found in Tab. B.1. The results of the calculations are shown in figure B.3 – droplet size distributions and figure B.4 – transmitted intensity signal.

Let’s have a look how the droplet size distribution evolves with time, see ﬁg- ure B.3. We ﬁnd that in the beginning most of the droplets are formed during the nucleation pulse, and the droplet size distribution has a sharp bell-like shape. Nev- ertheless, after the nucleation pulse, the number of droplets with smaller sizes increases. The widening of the droplet size distribution is caused by the appearance of new droplets. The formation of new nuclei indicates that the decrease of supersaturation after the nucleation pulse does not quench the nucleation process.

The predicted transmitted intensity signal coincides with the signal observed in the experiment. So, such a simple model can be used to interpret the nucleation and droplet growth data for methane/n-nonane mixtures.

Methane/water mixture

Using the available pressure and temperature profiles the droplet size distributions were calculated for the methane/water mixture, 03nov05_001 experiment. More details about experimental conditions can be found in Tab. B.1. The results of the calculations are shown in figure B.5 – droplet size distributions and figure B.6 – transmitted intensity signal.

After the nucleation pulse, the droplet size distribution has two maxima (see ﬁgure B.5, droplet size distribution 3). The larger maximum consists of droplets grown from nuclei produced in the nucleation pulse; the smaller maximum consists of droplets that have grown from nuclei produced in the pre-pulse period, where the water supersaturation is still high enough that a noticeable amount of nuclei appears in the system. With time this two-maximum distribution trans- forms to the asymmetrical bell-shape distribution with a distinct large-size tail, see ﬁgure B.5 and droplet size distribution 5. Moreover, the area under the distribution curve decreases with time, which means that the total amount of droplets is decreasing. Apparently, smaller droplets are evaporating and lager droplets are growing.

Analyzing the behavior of the droplet size distribution we conclude that at this experimental condition, two processes are essential. First, the high nucleation rate and high droplet growth rate causes the depletion of water vapor in a time interval comparable with the duration of the nucleation pulse. Second, the depletion of water vapor results in increasing the critical cluster size. The droplets with a size

123 APPENDIX B. SIMULTANEOUS NUCLEATION AND DROPLET GROWTH IN THE EXPANSION WAVE TUBE

11 10 Droplet size distribution 2 x 10 Droplet size distribution 1 x 10 12

10 6

8 5 ) ) 3 3 4 6 N (1/m N (1/m

2 1

0 0 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 −7 −7 r (m) x 10 r (m) x 10

12 12 x 10 Droplet size distribution 3 x 10 Droplet size distribution 4 16 14

10 10 ) ) 3 3 8 8 N (1/m N (1/m

6 6

4 4

2 2

0 0 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 −7 −7 r (m) x 10 r (m) x 10

12 x 10 Droplet size distribution 5

Pressure profile

45 12

3 44

80 10

42 8 ) 3

60 N (1/m

6 Pressure(bar)

50 4 6.0 6.5 7.0 7.5 8.0

5 4

40 2

0 5 10 15 20 25 30 35 40

0 Time (ms) 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 −7 r (m) x 10

Fig. B.3: Droplet size histograms and a pressure proﬁle for methane/n-nonane mixture. The conditions are the same as in 01nov05_001 experiment. On the pressure proﬁle picture the moments are indicated, at which the droplet size histograms are taken.

124 REFERENCES

1.00

0.75 0 /I

0.50

trans I

0.25

0.00

0 5 10 15 20 25 30 35 40

Time (ms)

Fig. B.4: Transmitted intensity signal for methane/n-nonane, 01nov05_001 experiment. The solid line is experimental data and the dashed line is model prediction. less than the critical size are unstable and these droplets have to evaporate. The evaporation of small droplets slightly increases the supersaturation and it allows the larger droplets to grow further. The second process, evaporation of small and growth of large droplets is signiﬁcantly slower than the ﬁrst process of nucleation and droplet growth.

These two processes with different time constants result in the appearance of a “kink” in the transmitted intensity signal, see figure B.6. This “kink” is also observed in the experimental transmitted signal. The experimental transmitted intensity signal somewhat differs from the signal predicted by model. Apparently, in the experiment the nucleation and droplet growth are going faster than in the model. In the model to calculate the nucleation rate we use a fit of experimental data. For conditions of this particular experiment 03nov01_001 the nucleation rate fit is extrapolated and this might lead to an underestimation of the nucleation rate.

References

[1] M.J.E.H. Muitjens, V.I. Kalikmanov, M.E.H. van Dongen, A. Hirschberg, and P.A.H. Derks. On mist formation in natural gas. Revue de l’Institut Français du Pétrole, 49(1):63–72, jan-feb 1994.

[2] P. Peeters. Nucleation and condensation in gas-vapor mixtures of alkanes and water.

125 APPENDIX B. SIMULTANEOUS NUCLEATION AND DROPLET GROWTH IN THE EXPANSION WAVE TUBE

12 13 x 10 Droplet size distribution 1 x 10 Droplet size distribution 2

4.5 5

4 4.5

4 3.5

3.5 3

3 ) ) 3 2.5 3 2.5 N (1/m N (1/m 2 2

1.5 1.5

1 1

0.5 0.5

0 0 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 −7 −7 r (m) x 10 r (m) x 10

15 14 Droplet size distribution 4 x 10 Droplet size distribution 3 x 10 3.5

12 3

10 2.5

8 ) ) 2 3 3 N (1/m N (1/m 6 1.5

4 1

0.5 2

0 0 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 −7 −7 r (m) x 10 r (m) x 10

14 x 10 Droplet size distribution 5

Pressure profile

4 90

45 2

1 3.5 3

70 2.5

42 ) 3

60 2 41 N (1/m Pressure(bar)

40 1.5 50

6.0 6.5 7.0 7.5 8.0

5 4 1

0.5

0 5 10 15 20 25 30 35 40

0 Time (ms) 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 −7 r (m) x 10

Fig. B.5: Droplet size histograms and a pressure proﬁle for methane/water mixture. The conditions are the same as in 03nov05_001 experiment. On the pressure proﬁle picture the moments are indicated, at which the droplet size histograms are taken.

126 REFERENCES

1.0

"kink"

0.9 0 /I

trans I

0.8

0.7

0 5 10 15 20 25 30 35 40

Time (ms)

Fig. B.6: Transmitted intensity signal for methane/water, 03nov05_001 experiment. The solid line is experimental data and the dashed line is model prediction.

127 APPENDIX B. SIMULTANEOUS NUCLEATION AND DROPLET GROWTH IN THE EXPANSION WAVE TUBE

PhD thesis, Eindhoven University of Technology, 2002.

128 Appendix C

Experimental data

exp Tsat (K) Psat (bar) Qc9, nl/min1 Qc1, nl/min Qc3, nl/min 11/8/2006-02 291.64 95.88 0.128 1.669 0.198 11/14/2006-01 291.66 95.88 0.111 2.010 0.236 11/14/2006-02 291.65 95.86 0.143 2.035 0.242 11/15/2006-01 291.64 95.88 0.175 2.056 0.247 11/15/2006-03 291.64 95.88 0.184 1.807 0.220 11/16/2006-01 291.66 95.86 0.186 1.992 0.242 11/16/2006-02 291.64 95.88 0.194 2.083 0.253 11/16/2006-03 291.67 95.88 0.122 1.936 0.228 11/17/2006-01 291.65 95.88 0.138 2.197 0.260 11/17/2006-02 291.64 95.88 0.170 1.844 0.871

Tab. C.1: Saturation conditions (Qc3 ﬂow is a mixture of methane/propane(0.1)).

1nl/min is the ﬂow in liter per minute measured at some reference point but converted to normal ◦ conditions 1.01325 bar absolute pressure and 0 C.

129 APPENDIX C. EXPERIMENTAL DATA

exp Tsat (K) Psat (bar) Qc9 (nl/min) Qc1 (nl/min) Qco2 (nl/min) 28-03-2006 001 290.25 53.10 0.833 0.284 0.379 28-03-2006 002 290.25 53.10 0.886 0.245 0.383 28-03-2006 003 290.24 53.10 0.915 0.199 0.382 29-03-2006 001 290.25 53.11 0.971 0.159 0.383 29-03-2006 002 290.25 53.10 1.087 0.126 0.419 29-03-2006 004 290.24 53.13 0.860 0.353 0.407 30-03-2006 001 290.26 53.10 0.824 0.402 0.414 06-04-2006 002 290.24 53.11 1.260 0.145 0.470 06-04-2006 003 290.24 53.11 1.212 0.198 0.472 08-04-2006 001 290.24 53.11 1.229 0.107 0.578 08-04-2006 002 290.24 53.15 1.178 0.102 0.553 10-04-2006 001 290.24 53.11 1.398 0.338 0.432 10-04-2006 003 290.23 53.13 1.376 0.332 0.426 11-04-2006 001 290.23 53.11 1.606 0.879 0.000 11-04-2006 002 290.22 53.11 1.604 0.879 0.000 11-04-2006 003 290.23 53.11 1.430 0.455 0.333 11-04-2006 004 290.22 53.11 1.417 0.451 0.329 12-04-2006 001 290.24 53.11 1.437 0.456 0.332 12-04-2006 002 290.24 53.08 1.297 0.000 0.702 12-04-2006 003 290.24 53.08 1.280 0.000 0.692

Tab. C.2: Saturation conditions.

exp Psat (bar) Tsat (K) Qc1(nl/min) Qc9 (nl/min) Qco2(nl/min) 01dec04, 001 59.99 290.67 0.844 0.845 0.420 02dec04, 001 59.99 290.66 0.877 0.877 0.437 02dec04, 002 59.97 290.66 1.006 0.796 0.452 03dec04, 001 59.95 290.66 1.260 0.779 0.511 03dec04, 002 59.97 290.67 1.233 0.863 0.527 04dec04, 001 59.95 290.66 1.243 1.101 0.587 04dec04, 002 59.95 290.66 0.899 1.010 0.477 07dec04, 001 59.97 290.65 0.668 0.756 0.355 07dec04, 002 59.97 290.67 0.788 0.702 0.373

Tab. C.3: Saturation conditions.

exp Psat (bar) Tsat (K) Qc1 (nl/min) Qc9 (nl/min0 Qco2 (nl/min) 14dec04, 003 40.01 290.66 0.000 2.264 0.571 14dec04, 002 40.00 290.66 0.103 2.148 0.563 14dec04, 001 40.01 290.66 0.103 2.150 0.564 15dec04, 001 40.00 290.66 0.000 2.262 0.570 15dec04, 002 40.01 290.66 0.056 2.247 0.526 16dec04, 001 40.01 290.66 0.147 2.118 0.567

Tab. C.4: Saturation conditions. Qco2 is a methane/carbon dioxide(0.05) mixture.

130 exp Psat (bar) Tsat (K) Qc1 (nl/min) Qc9 (nl/min) Qco2 (nl/min) 16nov04, 001 93.98 290.66 1.845 0.169 0.504 18nov04, 002 93.98 290.67 1.515 0.249 0.442 18nov04, 001 93.98 290.66 1.651 0.210 0.467 19nov04, 002 93.98 290.67 1.465 0.109 0.394 22nov04, 001 93.96 290.66 1.529 0.141 0.417 22nov04, 002 94.25 290.67 1.735 0.202 0.483 24nov04, 001 94.55 290.66 1.735 0.203 0.519 25nov04, 001 93.98 290.66 1.235 0.146 0.369 26nov04, 001 93.98 290.66 1.974 0.179 0.542 26nov04, 002 93.98 290.66 1.977 0.197 0.546

Tab. C.5: Saturation conditions. Qco2 is a methane/carbon dioxide(0.05) mixture.

131 APPENDIX C. EXPERIMENTAL DATA hs(a)Tp K cn K cn br c pm c ( J yc3 (ppm) yc9 (bar) Pcond (K) Tcond (K) Thps (bar) Phps 812602094. 0000040+604 2.1 1.2 4.7 3.8 0.47 5.5 0.40 4.2 6.8E+17 0.47 3.1 4.0E+16 0.36 0.030 1.2 4.3E+16 0.50 0.010 3.3 1.4E+17 0.45 0.010 71.4 5.1E+16 0.47 0.010 50.0 3.4E+17 0.48 0.010 50.1 6.4E+16 0.47 0.010 72.2 8.8E+16 0.43 39.6 0.010 72.1 3.4E+16 40.3 0.010 78.0 9.6E+16 40.0 0.010 66.4 40.3 240.3 0.010 55.6 40.4 240.9 44.4 40.0 240.0 60.1 295.9 40.5 240.4 296.0 40.3 240.6 295.3 40.1 240.1 87.8 295.3 40.1 240.5 88.1 295.4 240.4 87.8 295.7 239.8 87.9 295.2 239.6 88.0 295.5 88.1 295.1 88.1 295.9 88.0 87.9 88.4 a.C6 -oaenceto aedt o ehn/rpn mix methane/propane for data rate nucleation n-Nonane C.6: Tab. m − 3 s − 1 (ms) t ) tures. d r 2 / d t × 10 12 ( m 2 /s )

132 3 1 2 2 exp Phps (bar) Thps (K) Pcond (bar) Tcond (K) yc9 (ppm) yco2 J (1/m s− ) ∆t (ms) dr /dt (m /s) 28-03-2006 001 25.0 294.1 10.3 236.1 198.8 0.253 4.4E+15 0.45 14.3 28-03-2006 002 25.4 294.7 10.0 234.2 208.9 0.253 1.0E+17 0.43 12.4 28-03-2006 003 25.4 295.1 10.1 235.1 218.2 0.255 6.6E+16 0.44 13.6 29-03-2006 001 25.2 294.4 10.3 235.8 229.1 0.253 3.4E+16 0.45 14.0 29-03-2006 002 25.5 295.2 10.0 234.5 237.6 0.257 2.4E+17 0.46 0.0 29-03-2006 004 25.6 295.6 10.0 234.5 189.4 0.251 9.2E+15 0.42 12.2 30-03-2006 001 25.4 294.9 10.2 235.6 179.4 0.252 7.2E+14 0.44 12.5 06-04-2006 002 25.6 295.4 10.1 235.1 239.8 0.250 3.4E+17 0.45 06-04-2006 003 25.6 295.4 10.1 234.8 229.8 0.251 5.7E+16 0.47 12.9 08-04-2006 001 25.5 294.1 10.3 234.8 229.1 0.302 4.4E+16 0.45 13.6 08-04-2006 002 25.5 294.1 10.2 234.1 229.4 0.302 9.2E+16 0.43 14.4 10-04-2006 001 25.3 294.6 10.1 234.9 230.1 0.199 4.5E+16 0.44 14.9 10-04-2006 003 25.5 295.4 10.2 235.3 230.1 0.199 1.1E+16 0.45 16.2 11-04-2006 001 25.2 295.0 10.0 234.6 230.5 0.000 3.1E+15 0.40 18.8 11-04-2006 002 25.3 295.3 10.2 235.3 230.2 0.000 9.0E+16 0.42 19.2 11-04-2006 003 25.4 295.6 10.3 236.3 229.8 0.150 1.2E+15 0.45 17.5 11-04-2006 004 25.4 295.3 9.9 234.1 230.0 0.150 1.4E+16 0.46 14.9 12-04-2006 001 25.1 294.6 10.2 235.7 230.4 0.149 2.1E+15 0.34 17.2 12-04-2006 002 25.7 295.6 10.1 234.6 231.4 0.351 3.4E+16 0.44 13.3 12-04-2006 003 25.8 296.0 9.9 233.8 231.5 0.351 8.3E+16 0.45 13.4

Tab. C.7: n-Nonane nucleation rate data for methane/carbon dioxide mixtures. 133 APPENDIX C. EXPERIMENTAL DATA 7e0,022535. 532001000005.6E+15 4.6E+16 0.010 1.9E+17 0.009 1.6E+15 160.0 0.010 9.6E+14 180.2 0.010 1.9E+14 179.6 0.010 3.3E+15 240.0 159.4 0.010 2.3E+16 239.7 139.7 0.010 2.5E+16 238.0 129.6 0.009 240.1 149.9 0.009 25.3 239.5 169.9 25.2 239.5 170.3 24.5 239.3 25.4 56.3 239.3 25.1 55.6 239.3 25.1 54.3 24.9 295.3 54.3 25.0 294.5 55.8 25.0 002 07dec04, 292.6 54.6 001 07dec04, 292.5 55.9 002 04dec04, 294.9 55.2 001 04dec04, 293.1 55.6 002 03dec04, 295.0 001 03dec04, 294.0 002 02dec04, 294.5 001 02dec04, 001 01dec04, x hs()Pp br cn br cn K c pm c2J( J yco2 (ppm) yc9 (K) Tcond (bar) Pcond (bar) Phps (K) Thps exp a.C8 -oaenceto aedt o ehn/abndioxi methane/carbon for data rate nucleation n-Nonane C.8: Tab. emixtures. de m − 3 s − 1 )

134 3 1 exp Thps (K) Phps (bar) Pcond (bar) Tcond (K) yc9 (ppm) yco2 J (m− s− ) 14dec04, 003 293.8 25.0 10.0 233.3 261.4 0.010 2.6E+16 14dec04, 002 295.0 25.3 10.1 234.5 249.8 0.010 2.3E+16 14dec04, 001 293.9 25.1 10.9 233.5 249.9 0.010 3.4E+15 15dec04, 001 294.1 25.2 10.0 234.1 261.4 0.010 2.0E+14 15dec04, 002 294.8 25.3 10.0 234.0 260.1 0.009 9.6E+14 16dec04, 001 294.8 25.3 10.0 233.9 244.9 0.010 1.7E+15

Tab. C.9: n-Nonane nucleation in methane/carbon dioxide mixture. 135 APPENDIX C. EXPERIMENTAL DATA 6o0,022518. 972906. .1 1.2E+16 8.8E+15 8.7E+15 0.010 2.2E+16 0.010 2.1E+16 0.011 62.7 6.0E+15 0.011 57.4 1.4E+16 0.010 72.2 2.2E+16 0.010 72.4 239.0 6.7E+16 0.010 72.9 239.6 1.8E+16 0.010 58.5 239.3 0.010 48.1 239.5 0.010 39.7 78.0 238.7 40.1 97.8 239.1 40.0 58.1 238.3 40.0 240.1 87.8 39.8 240.0 87.6 40.1 239.6 87.6 39.2 295.1 87.7 40.3 294.8 87.8 40.2 002 26nov04, 294.7 87.4 39.9 001 26nov04, 295.0 88.5 001 25nov04, 294.5 88.8 001 24nov04, 294.1 88.5 002 22nov04, 295.7 88.6 001 22nov04, 296.1 002 19nov04, 295.9 001 18nov04, 296.0 002 18nov04, 001 16nov04, x hs()Pp br cn br cn br c pm yco2 (ppm) yc9 (bar) Tcond (bar) Pcond (bar) Phps (K) Thps exp a.C1:nNnn ulainrt aafrmtaecro diox methane/carbon for data rate nucleation n-Nonane C.10: Tab. d mixtures. ide 1/( J m − 3 s − 1)

136 Summary

Nucleation of n-nonane in mixtures of methane, propane, and carbon dioxide

Understanding droplet formation and droplet growth processes is important for natural gas industry, in particular for the separation of condensable components. Natural gas is a complex gas-vapor mixture which contains methane and many other non-condensable and condensable components. The non-condensable components can appreciably inﬂuence droplet formation which has to be taken into account in the design and development of natural gas separators.

The driving force for condensation in a gas-vapor mixture is the deviation of the thermodynamic state from equilibrium, often expressed in terms of the vapor supersaturation. For a dilute vapor the supersaturation becomes the ratio of a vapor molar concentration to its value at phase equilibrium. In general, the condensation process can be split into two parts: (1) nucleation, formation of nuclei of the liquid phase, and (2) droplet growth. During nucleation, clusters of the liquid phase are mainly formed as a result of cluster-molecule interactions. To become a growing droplet, a cluster has to overcome an energy barrier. Clusters with a maximum in formation energy are called critical clusters. The formation rate of critical clusters is of crucial importance for the whole process of nucleation. The rate of formation of droplets, the nucleation rate, strongly depends on supersaturation, temperature, and pressure.

Nucleation and droplet growth have been realized experimentally in an expansion wave tube, in which a gas-vapor mixture is brought in a supersaturated state by means of a fast isentropic expansion. In the expansion tube, the nucleation pulse method is implemented; therefore nucleation and droplet growth processes are separated in time. Nucleation takes place at known and controlled conditions: temperature, pressure, and supersaturation. During this study several changes have been made in the original setup. The quality of the pressure measurements has been improved by simultaneously using two pressure transducers with different frequency characteristics. In this way, in situ calibration of the pressure Summary transducers has become possible. Furthermore, the mixture preparation part of the setup had been extended so that it is possible to study mixtures of one vapor and two carrier gas components.

The nucleation rate data are obtained in the form of nucleation isotherms: the supersaturation is varied at constant temperature and pressure. The nucleation isotherms can be analyzed with the nucleation theorem – a powerful tool to deduce critical cluster properties directly from experimental data. The nucleation theorem has been reformulated such that the size and the composition of the critical cluster are deduced from experimental data and from well-known phase equilibrium properties of the gas phase.

In order to study the inﬂuence of the non-condensable gases on condensation, the following model mixtures have been investigated: methane/n-nonane, methane/propane/n-nonane and methane/carbon dioxide/n-nonane. In these mixtures, n-nonane plays the role of a condensable heavy hydrocarbon.

The systematic investigation of n-nonane nucleation in methane has been continued. An additional series of experiments with n-nonane/methane mixtures has been performed. The experimental data reveals a strong inﬂuence of the methane carrier gas on nucleation: an increase of pressure facilitates nucleation, and alters the composition of the critical cluster. The numbers of methane and n-nonane molecules in the critical cluster have been evaluated by applying the nucleation theorem. For high pressures (>10 bar) the cluster composition differs substantially from the composition of the equilibrium liquid phase of the methane/n-nonane mixture at the same conditions. Evidently, the capillarity approximation, widely used in nucleation theories, cannot be applied to the methane/n-nonane system for pressures above 10 bar.

A series of experiments has been performed in methane/propane carrier gas mixtures with different concentrations of propane. The data was obtained at conditions at which they can be compared with available data for methane/n-nonane mixtures. It is found that at 0.01-0.03 molar concentrations of propane and at high pressures (40 bar), propane inﬂuences n-nonane nucleation: the nucleation rate is increased about 5 times by the presence of propane in methane/propane (0.01) for the same value of the n-nonane supersaturation. This shows, that propane substi- tutes some methane molecules in the critical cluster, thereby lowering the energy of critical cluster formation.

Experimental data has been obtained for n-nonane nucleation in binary methane/carbon dioxide mixtures at different temperatures and pressures. A comparison with the methane/n-nonane data shows that carbon dioxide inﬂuences nucleation when its molar concentration in the mixture exceeds 0.25. At 235 K and 10 bar the n-nonane nucleation rate in methane/carbon dioxide (0.25) is about one

138 Summary order of magnitude higher than the n-nonane nucleation rate in pure methane at the same value of supersaturation. The addition of carbon dioxide to the carrier gas not only changes the equilibrium state, but it also changes the energy of critical cluster formation.

A method is proposed to estimate the nucleation rate of a dilute vapor (molar 4 fraction 10− ) in the presence of a carrier gas mixture. Because of the abundantly present carrier gas molecules, a dynamic equilibrium for the interactions of clusters with carrier gas molecules is achieved; as a result, the kinetic model for multicomponent nucleation can be reduced to a quasi-unary one. In the quasi-unary expression for the nucleation rate, the material properties of the critical cluster are assumed to be the same as the material properties of the corresponding equilibrium liquid mixture. The comparison with experimental rate data for methane/n- nonane shows that the quasi-unary theory yields nucleation rates comparable with experimental ones at rather low pressures (10 bar). But at high pressures (40 bar) the theory neither predicts the nucleation rate nor the composition of the critical cluster correctly. This is consistent with the experimental observation that the critical cluster composition at high pressure substantially differs from the equilibrium composition of the corresponding equilibrium liquid mixture.

The gradient theory of the phase interface offers an approach to determine density proﬁles of mixture components in the interface and the energy of surface formation, if the bulk thermodynamic properties are known for both phases at equilibrium. This theory can also be applied to a critical cluster which is in a metastable equilibrium. It yields the size, the structure and the energy of formation of the critical cluster. Results from the gradient theory are used to calculate the nucleation rates for pure n-nonane. The gradient theory appears to predict the properties of the critical clusters qualitatively well.

139 Summary

140 Samenvatting

Het begrijpen van druppelvorming en van druppelgroei is belangrijk voor de aard- gasindustrie, in het bijzonder voor de scheiding van condenseerbare componenten. Aardgas is een complex mengsel van methaan en talrijke andere condenseerbare en niet-condenseerbare componenten. De niet-condenseerbare componenten kunnen het druppelvormingsproces aanzienlijk beïnvloeden, hetgeen van belang is voor het ontwerp en de ontwikkeling van condensaatscheiders voor aardgas.

De drijvende kracht voor condensatie in een gas-dampmengsel is de afwijking van thermodynamisch evenwicht, vaak uitgedrukt in termen van dampoververza- diging. Voor een verdunde damp wordt de oververzadiging gelijk aan de ver- houding van de damp-molfractie en de waarde daarvan bij fase-evenwicht. In het algemeen kan het condensatieproces worden opgesplitst in twee delen: (1) nucleatie, de vorming van vloeistofkernen, en (2) druppelgroei. Tijdens nucleatie worden vloeistofclusters hoofdzakelijk gevormd door cluster-molecuul interactie. Om een macroscopische druppel te worden moet een cluster eerst een energiebarriëre overwinnen. Clusters met een maximale vormingsenergie worden kritische clusters genoemd. De vormingssnelheid van kritische clusters is bepalend voor het gehele nucleatieproces. De vormingssnelheid van druppels, de nucleatiesnelheid, is sterk afhankelijk van oververzadiging, temperatuur en druk.

Nucleatie en druppelgroei zijn experimenteel gerealiseerd in een expansiegolf- buis, waarin een gas-dampmengsel in een oververzadigde toestand wordt gebracht door middel van een snelle isentrope expansie. De expansiebuis maakt gebruik van het nucleatiepulsprincipe, hetgeen inhoudt dat nucleatie en druppelgroei zijn gescheiden in de tijd. Nucleatie vindt plaats bij een bekende en gecontroleerde toestand van temperatuur, druk, en samenstelling. Tijdens dit onderzoek zijn een aantal veranderingen aangebracht in de experimentele opstelling. De kwaliteit van de drukmetingen is verbeterd door het gelijktijdig gebruik van twee typen druksensoren met verschillende frequentiekarakteristieken. Daarmee werd in situ calibratie van de druksensoren mogelijk. Verder werd de gasmengselpreparator uitgebreid zodanig dat een bestudering van mengsels bestaande uit een damp en twee verschillende gassen mogelijk werd. Samenvatting

De experimentele nucleatiesnelheden zijn bepaald in de vorm van nucleatie-isothermen: de oververzadiging werd gevarieerd bij constante temperatuur en druk. De nucleatie-isothermen kunnen worden geanalyseerd met het nucleatietheorema – een krachtig hulpmiddel om de eigenschappen van kritische clusters af te lei- den uit experimentele nucleatiegegevens. Het nucleatietheorema werd in een zo- danige vorm gebracht dat de grootte en de samenstelling van kritische clusters volgen uit nucleatiesnelheden en bekende eigenschappen van de damp in fase- evenwicht.

De systematische studie van n-nonaan-nucleatie in methaan is verder voortgezet. Een additionele reeks experimenten is uitgevoerd. De experimentele gegevens bevestigen de sterke invloed van het methaan op n-nonaan nucleatie: een toename van de druk bevordert nucleatie en verandert de samenstelling van het kritische cluster. De aantallen methaan- en n-nonaanmoleculen in een kritisch cluster zijn bepaald door toepassing van het nucleatietheorema. Bij hoge druk (>10 bar) verschilt de clustersamenstelling aanzienlijk van de samenstelling bij fase-evenwicht bij dezelfde temperatuur en druk. Het is duidelijk dat de "capillarity approximation", die veelvuldig wordt toegepast in de beschrijving van nucleatie, niet zinvol is voor het methaan/n-nonaan systeem bij drukken groter dan 10 bar.

Een reeks experimenten is uitgevoerd in mengsels van methaan/propaan/n-nonaan, voor verschillende concentaties van n-nonaan. De experimentele resultaten werden vergeleken met nucleatiedata voor methaan/n-nonaan-mengsels. Een 0,01- 0,03 molaire concentratie propaan heeft een duidelijke invloed op de nucleatiesnelheid bij een relatief hoge druk (40 bar): de nucleatiesnelheid wordt door de aanwezigheid van propaan (0,01 molfractie) ongeveer een factor 5 vergroot, voor dezelfde waarde van de n-nonaan-oververzadiging. Dit laat zien dat in een kritisch cluster de methaanmoleculen voor een deel worden vervangen door propaanmole- culen, waarbij de vormingsenergie van kritische clusters wordt verlaagd.

Experimentele gegevens zijn verkregen voor n-nonaan-nucleatie in binaire methaan/kooldioxide-mengsels bij verschillende temperaturen en drukken. Een vergelijking met de methaan/n-nonaan-experimenten leert dat kooldioxide het nucleatieproces van n-nonaan beïnvloedt voor molaire molfracties van kooldioxide groter dan 0,25. Bij 235 K en 10 bar is de nucleatiesnelheid van n-nonaan in aanwezigheid van kooldioxide een orde groter dan zonder kooldioxide, bij dezelfde oververzadiging van n-nonaan. De toevoeging van kooldioxide aan het gasmeng- sel beïnvloedt niet alleen de evenwichtssamenstelling, maar ook de vormingsenergie van kritische clusters.

Een methode wordt beschreven om de nucleatiesnelheid van een verdunde damp 4 (molfractie 10− ) in de aanwezigheid van een draaggas te berekenen. Omdat de draaggasmoleculen overvloedig aanwezig zijn, is er een dynamisch evenwicht van clusters en gasmoleculen. Als gevolg daarvan kan het kinetische model voor

142 Samenvatting multicomponentnucleatie in dit geval worden gereduceerd tot een quasi-unair model. In de quasi-unaire uitdrukking voor de nucleatiesnelheid worden materiaal- gegevens gesubstitueerd zoals die bekend zijn bij fase-evenwicht. Een vergelijking met experimentele nucleatiesnelheden voor n-nonaan in methaan laat een redelijke overeenkomst zien voor drukken beneden 10 bar. Bij hogere druk echter (40 bar) wordt noch de nucleatiesnelheid correct voorspeld, noch de samenstelling van het kritische cluster. Dit is consistent met de experimentele waarneming dat bij deze druk de samenstelling van het kritische cluster aanzienlijk verschilt van de samenstelling bij fase-evenwicht.

De gradiënttheorie van het fase-oppervlak biedt een mogelijkheid om zowel dicht- heidsproﬁelen van het fase-oppervlak te bepalen als de oppervlakte-energie, uit- gaande van bekende thermodynamische eigenschappen van de verschillende componenten. Deze theorie kan ook worden toegepast op een kritisch cluster in meta- stabiel evenwicht. De theorie geeft de grootte, de ruimtelijke verdeling van de verschillende componenten en de energie van het kritische cluster. Daarmee kan ook de nucleatiesnelheid worden berekend. Met de gradiënttheorie is de nucleatiesnelheid van zuiver n-nonaan berekend. De theorie geeft een kwalitatief correcte beschrijving van de eigenschappen van kritische clusters.

143 Samenvatting

144 Acknowledgments

During the preparation of the thesis, many people have made contributions, directly or indirectly, and I am very grateful to all of them.

I am very grateful to my supervisor prof. Rini van Dongen for providing me the possibility to study and work at TUE. He has a rare gift to encourage and motivate people around him and I was no exception. It was a great pleasure for me to work under his supervision. I would like to thank my second supervisor, Prof. Mico Hirschberg for giving his critical but very friendly suggestions during the development of this study.

I would like to express my sincere appreciation to dr. Jan Hrubý for the development of gradient theory and for his carefully commenting on the manuscript. I am very much indebted to dr. Vitaly Kalikmanov and to dr. Sergey Fisenko for very useful and enjoyable discussions on nucleation and other issues, and for their support. The project has been partially supported by Twister B.V., partially by STW. I did appreciate the encouragement and interest expressed by Marco Betting and Bart Prast of Twister B.V. and of the other members of the user committee: John Janssen (Gasunie) and prof. Michael Golombok (Shell).

The experimental study demands a lot of efforts, and I very much appreciate help and support of the technicians: Jan Willems, Ad Holten, Herman Koolmees, and Feek van Uittert. Also many thanks go to Petr Krejˇcí and Peter Jonkers for assisting me in the laboratory.

Further, I wish to thank Vincent Holten, my ofﬁce mate, for his interest in my work and for valuable remarks and suggestions.

I thank the complete staff of the “Fluid Dynamic Laboratory”, especially Marjan Pepers, who created a friendly and hospitable environment.

Finally, my deepest gratitude to my family and Zhenya for their support and love. Acknowledgments

146 Curriculum Vitae

13 June 1978 : born in Smolevichi, Belarus

1995 - 2000 : Belarusian State University (Minsk), Diploma with honors in physics, Thesis: “Conversion of methane into hydrogen at conditions of superadiabatic catalytic process of partial oxidation”

2000 - 2002 : engineer-physicist at Lab. Chem. and Phys. of Combustion, Heat and Mass Transfer Institute (Minsk)

2002 - 2007 : PhD research at the Gas Dynamics Group, Eindhoven University of Technology