J. Phys. Chem. B 2001, 105, 11763-11771
11763
High Pressure Nucleation Experiments in Binary and Ternary Mixtures† Paul Peeters,*,‡ Jan Hruby´ ,§ and Marinus E. H. van Dongen‡ Department of Applied Physics, EindhoVen UniVersity of Technology, P.O. Box 513, 5600 MB EindhoVen, The Netherlands, and Institute of Thermomechanics, Academy of Sciences of the Czech Republic, DolejsˇkoVa 5, CZ-182 00 Prague 8, Czech Republic ReceiVed: May 1, 2001; In Final Form: August 16, 2001
New experimental high-pressure nucleation results of n-nonane in methane and first nucleation rate data of water in methane are presented. The nucleation behavior of the ternary system water and n-nonane in methane is investigated experimentally. For this purpose, a new dedicated mixture preparation device was designed, rendering more accurate and controllable vapor fractions. This has eliminated the large scatter in the nucleation rates of n-nonane in methane at 40 bar and 240 K, previously measured. These experiments also show that nucleation theories are unable to predict the rates (even qualitatively) at these extreme conditions. The composition of the critical cluster is determined for all of the binary systems investigated, by applying the nucleation theorem. Furthermore, a criterion for the application of one component theories to binary gasvapor systems is applied. The nucleation data for the ternary mixtures are analyzed using the experimental results for the binary mixtures. From this, it can be concluded that water and n-nonane nucleate independently in mixtures of water, n-nonane, and methane.
1. Introduction Nucleation from the vapor phase and subsequent droplet growth at high pressure is of particular interest for the natural gas industry. Nucleation and growth can be used to separate vapor components (water and heavy hydrocarbons) from the gas (mainly methane). Optimization of newly developed separators depends to a large extent on our knowledge of homogeneous nucleation and droplet growth processes in high pressure mixtures of methane, higher alkanes, and water. Such knowledge is still rather limited,1,2 if it exists at all. In our laboratory, nucleation and growth experiments are performed using a pulse expansion wave tube. This is basically a shock tube, the high pressure section (HPS) of which is used as the test section. The advantage of this setup is that it can be operated at pressures (up to 100 bar) commonly encountered in natural gas industry. So far, several binary gas-vapor mixtures have been studied, including mixtures of methane and n-nonane and methane and n-octane3-5. Besides, a study of nucleation and growth in actual (dry, i.e., without water) natural gas has been performed.6 One of the major challenges when designing and performing nucleation experiments is to accurately control the composition of the gas-vapor mixture. Vapor fractions are of the order of 10-5-10-4 and have to be known within an accuracy of a few percent to do meaningful experiments. With the previous setup,5 a binary gas-vapor mixture was inserted into the HPS, and then its hydrocarbon or water content was measured just before the actual experiment. Because natural gas contains many vapor components, it was desirable to be able to do experiments with mixtures containing more than one vapor component, in a controlled manner. Therefore,
a new mixture preparation device was designed. With this setup, the HPS can be filled with a ternary gas-vapor-vapor mixture. The composition is determined a priori on the basis of known binary phase equilibrium data. 2. Nucleation in Dilute Gas-Vapor Mixtures We shall now concentrate on gaseous mixtures consisting of one or more vapors and one single (supercritical) background gas. Real gas effects are important. The nucleation rate J will depend on the total pressure and temperature, and the supersaturation of the r vapor components J(p, T, S1, S2, ..., Sr). The supersaturation indicates how far from equilibrium the system is and is defined here as
Si ) exp
(
)
µV,i(p, T, y1, y2, ..., yr) - µV,i(p, T, 0, 0, ..., yieq, ..., 0) (1) kBT
The first (left) chemical potential of vapor i is taken at the actual (supersaturated) thermodynamic state, at which the nucleation takes place. The second (reference) chemical potential is taken at a state where all but the ith vapor fractions are zero and component i is at its dew point. When the background gas is abundant, and hence all of the vapor fractions are small, only gas-gas and vapor-gas interactions have to be considered. In that case, eq 1 simplifies to
Si )
yi yieq
(2)
†
Part of the special issue “Howard Reiss Festschrift”. * To whom correspondence should be addressed. Fax: (0)40 2464151. E-mail:
[email protected]. E-mail:
[email protected]. E-mail:
[email protected]. ‡ Eindhoven University of Technology. § Academy of Sciences of the Czech Republic.
where we have assumed that the fugacity coefficients of the supersaturated vapors do not differ from their values at phase equilibrium. The equilibrium molar vapor fraction can be expressed as7
10.1021/jp011670+ CCC: $20.00 © 2001 American Chemical Society Published on Web 10/12/2001
11764 J. Phys. Chem. B, Vol. 105, No. 47, 2001
pis(T) fe,i(p, T) yi (p, T) ) p eq
Peeters et al
(3)
where pis is the pure component saturated vapor pressure and fe,i is the vapor pressure enhancement factor. An important tool for the interpretation of nucleation experiments is the nucleation theorem, originally introduced by Kashchiev and later extended by Viisanen et al.8 and Oxtoby and Kashchiev.9 The nucleation theorem relates the partial derivatives of the nucleation rates with respect to the chemical potential of the different components to the composition of the critical cluster. The critical cluster is in meta-stable equilibrium at the given (supersaturated) conditions and plays an essential role in all nucleation theories. Therefore, application of the theorem to the experiments helps to improve theories and theoretical understanding. Luijten et al. have applied the nucleation theorem to a binary dilute gas-vapor mixture.5 They found
∂ ln J ) n/V + 1 ∂ ln S p,T
[
]
S,T
/ ) -n/Vxeq g + ngZg + 2 ln fe
∆G ˆ ) -Vl∆p + σA + nv∆µˆ v + ng∆µˆ g
(5)
where n* denote the number of molecules in the critical cluster (vapor or gas), xeq is the equilibrium bulk liquid fraction, and Zg is the gas compressibility factor. These relations will be used when analyzing the nucleation results of the binary systems. Furthermore, three different theories will be used to compare the new binary results to. One of these is the binary classical nucleation theory (BCNT), originally formulated by Reiss,10 and later extended by Staufer11 and by Wilemski.12 The ReissStaufer-Wilemski theory was applied by Looijmans to the highly nonideal mixture of methane and n-nonane.1 A different approach was followed by Kalikmanov and Van Dongen,13who modified the semiphenomenological Dillmann-Meier model.14 Later, the same authors extended the model to describe multicomponent nucleation.2 This model will be referred to as SPMT (semi-phenomenological multicomponent theory). Often, experimental nucleation rates of a vapor in a carrier gas are compared to single component theories. In the next section, we will discus the applicability of one component theories to these kind of systems. 3. Quasi-One-Component Nucleation Luijten and Van Dongen15 argued that nucleation in binary mixtures consisting of a carrier gas and a dilute vapor can in some cases be described with a one component theory. They derived a criterion for a one component theory to be adequate. Here we will follow their derivation and formulate their condition in a somewhat different way. Consider two binary systems, at the same temperature T and pressure p. In one of the systems, a single cluster is present, the other consists of monomers only. The difference in the Gibbs free energy of the two systems can then be written as
∆G ˆ ) -Vl(pl - p) + σA + nlv(µlv - µgv) + nlg(µlg - µgg) + nsv(µsv - µgv) + nsg(µsg - µgg) (6) The superscripts l, g, and s represent the liquid phase, gas phase, and the surface region. The subscripts v and g denote the vapor
(7)
Here, ∆µˆ i is the difference in chemical potential of a cluster with respect to the gas phase, ∆µˆ i ) µli(p + ∆p, xˆ i) - µgi (p, yi). So, for the gas component it reads
( )
∆µˆ g ) Vlg∆p + kBT ln
γˆ gxˆ g
(8)
eq γeq g xg
and for the (supersaturated) vapor component it reads
∆µˆ v ) Vlv∆p + kBT ln
(4)
and
[∂∂ lnln pJ]
and gas component, respectively. Furthermore, Vl is the volume of the cluster, σ is the surface tension, A is the surface area of the cluster, and µ is the chemical potential. When we assume that the liquid and the surface are in thermodynamic equilibrium, we have µsi ) µli, and we can lump the bulk molecules of the liquid and surface molecules together, ni ) nsi + nli, rendering
( ) γˆ vxˆ v
eq γeq v xv
- kBT ln(Sv)
(9)
Here, xˆ refers to the composition of the bulk of the cluster, whereas xeq refers to the equilibrium bulk liquid composition, and γ is the activity coefficient. Clusters of critical size are in unstable equilibrium with the rest of the system, and therefore, the ∆µˆ i’s are zero, rendering the well-known Kelvin relations. Now, an important assumption is made. Because we are always considering dilute vapors, it is assumed that the gas component in the gas phase is in equilibrium with the gas component in the cluster, for all cluster sizes. In that case, ∆µˆ g is always zero. One can now proceed in two ways. One is to determine ∆G ˆ (eq 7), substituting both eqs 8 and 9. This was done by Luijten and Van Dongen,15 and then all of the ∆p terms cancel (because Vl ) ngVlg + nvVlv). Another way to proceed is to leave out the ∆µˆ gterm in eq 7 directly. This results in
∆G ˆ ) σ4πr2 - nvkBT ln(Sv) - ngVlg
( )
γˆ vxˆ v 2σ + nvkBT ln eq eq r γv xv
(10) where ∆p ) 2σ/r was substituted and r is the radius of the cluster. The first two terms on the right-hand side of eq 10 represent the classical one component result. So, whenever the sum of the first two terms is much larger than the sum of the second two terms, a quasi-one-component theory can be used to determine the nucleation rate in a system consisting of a carrier gas plus a vapor. To check this minimum criterion, one has to have an expression for the surface tension. However, this can be avoided in the following way. For the critical cluster, the first two terms do not cancel and are of equal order, because we know from classical (one component) theory that ∆G* ) (1/3)σA. Therefore, we will now compare the first and third term and the second and fourth term. Using ngVlg + nvVlv ) 4/3πr3, this results in the following set of sufficient conditions:
ngVlg 2 , 1, 3 n Vl + n Vl g g V V
(11)
( )
(12)
and
|ln
γˆ vxˆ v
eq γeq v xv
|/ln(Sv) , 1
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J. Phys. Chem. B, Vol. 105, No. 47, 2001 11765
Now, the two contributions of the carrier gas can be evaluated separately. These contributions are the change in density of the critical cluster due to the presence of the carrier gas and the importance of the change in the chemical potential due to the entropy of mixing. When the criteria stated above are fulfilled, application of the classical (one component) nucleation theory (CNT) should give the same results as the BCNT. The CNT expression reads
(
JCNT ) KS2 exp -
)
4 θ3 27 (ln S)2
(13)
in which the pre-factor K and the dimensionless surface tension θ are mixture specific and can be expressed as functions of the pressure and temperature. They are specified in Appendix A. Since the first introduction of the CNT, many modifications have been suggested. One of these (now often used) is the internally consistent classical theory (ICCT):
(
JICCT ) KS exp θ -
)
4 θ3 27 (ln S)2
(14)
The factor 1/S results from the fact that detailed balance is applied at phase equilibrium, whereas the extra term θ in the exponential results from the requirement that the expression for the work of formation of an n-mer should hold down to monomers. 4. Experiment The setup developed in our laboratory is a wave tube, operating according to the nucleation pulse principle.4,16 The gas is adiabatically expanded during a short time interval in which nucleation occurs. Then, the nucleation process is stopped by a slight recompression to a less supersaturated state, such that the droplets can grow to an optically detectable size. 4.1. Wave Tube and Optics. The wave tube is basically a shock tube and consists of a 6.42 m long low pressure section (LPS) and a 1.25 m long HPS. The sections have an inner diameter of 36 mm and are separated by a polyester diaphragm before the start of an experiment. The nucleation pulse is formed by means of a local widening in the LPS, just behind the diaphragm. After the diaphragm has opened, reflections of the shock wave at the local widening will cause a small pressure dip at the end wall of the HPS. The whole process being adiabatic, this will also result in a small dip in the temperature. Because equilibrium vapor fractions are exponentially dependent on temperature, a large peak in the saturation will occur, i.e., the nucleation pulse. All of the droplets are formed during the very short nucleation pulse (typically 0.3 ms) and will all have the same size as they grow. Therefore, using a laser and measuring the light scattered by the cloud of droplets and the transmission through the cloud of droplets simultaneously, we can determine the radius of the droplets and their number density. The nucleation rate follows simply from the ratio of the number density to the duration of the nucleation pulse. With this setup, nucleation rates between 1013 and 1017 m-3s-1 can be measured at temperatures ranging from 220 to 260 K and pressures up to 50 bar. For more details the reader is referred to previous publications.4,16 4.2. Mixture Preparation. Nucleation rates are extremely dependent on the saturation ratio(s) of the vapor component(s). Hence, when doing experiments, small changes in the composi-
Figure 1. Schematic view of the MPD.
tion of the gas-vapor mixture can result in large changes in the nucleation rates. Keeping the errors in the composition as small as possible is therefore of key importance. To achieve this, a new mixture preparation device (MPD) was designed. It is schematically shown in Figure 1. The MPD can be split into several sections, going to continuously lower levels of operating pressure. First, the dry carrier gas comes in at a maximum pressure of 100 bar. Then, the gas stream is split into three different branches, each of them controlled by a mass flow controller (MFC). Two gas streams are saturated with a vapor component at pressure psat (maximum of 95% of the upstream pressure) of the saturation section and a temperature somewhat lower than ambient. Saturation is established by bubbling the dry gas through two vessels containing the liquids to be vaporized.5 The third gas stream consists of the dry gas, with which the other two can be further diluted. The flows are controlled by Brooks MFCs with different ranges. The dilution flow has a maximum rate of 3000 sccm N2. MFCs 1 and 2 can have maximum rates of 300, 1500, or 3000 sccm N2. When gases other than nitrogen are used (as in our case), it is best to recalibrate the MFCs, to maintain the highest possible accuracy. Therefore, a Brooks Vol-U-Meter Gascalibrator is available, enabling us to calibrate the MFCs with different gases and different upstream pressures. The actual composition of the gas mixture leaving the saturation section is determined by three factors, those being the temperature and pressure at which saturation takes place and the ratio’s of the three flows. Hence, by controlling these factors, the composition of the mixture can be controlled. The temperature of the saturators is held fixed by a thermostatic bath. The pressure and the ratio’s of the flows are controlled by a combined control circuit. As the gas mixture leaves the saturation section through the needle valve in the heated box, the pressure in this section could decrease. The actual pressure is continuously measured and compared with a set point. When the actual pressure falls below the set point, the flow of one of the MFCs (MFC 0 in Figure 1) is increased. At the same time, the flows of the other (operating) MFCs are also increased in such a way that the ratio of the flows remains constant. The last section goes from the needle valve in the high temperature box to the waste outlet and includes the HPS of the wave tube. Prior to each experiment, this section is evacuated, and therefore, the pressure here changes from 0 bar up to the desired initial pressure in the HPS. The gas mixture leaving the saturation section through the needle valve can experience a large pressure drop (maximum 95 bar) and a correspondingly large temperature drop, due to the JouleThomson effect. To prevent premature condensation, the needle
11766 J. Phys. Chem. B, Vol. 105, No. 47, 2001 valve is placed in a box that can be heated to 200 °C. Also placed in this box is a mixing vessel to ensure that the gas mixture is homogeneous. The direction of the flow is indicated by the arrows in Figure 1. The gas mixture first enters the HPS near the end wall and leaves it again near the diaphragm. As the pressure increases, it is continuously measured and compared to the set point (the desired initial pressure in the HPS). Once this set point has been reached, the upstream pressure controller (UPC) will (slightly) open a valve. In this way, the HPS is continuously flushed at constant pressure. Flushing is necessary because thermodynamic equilibrium needs to be established between the walls of the setup and the gas mixture itself. During flushing, the vapor content can be monitored using a humidity sensor (RH) and a gas chromatograph (GC). Typically, the volume of the HPS needs to be replaced three times before the composition leaving the HPS takes a constant value. As an extra check, the flow can be redirected in such a way that the composition of the gas mixture is measured before it enters the HPS. It should be noted here that the GC and RH are only used as relative indicators and that the actual composition is calculated from the conditions in the saturation section. This has proven to give very consistent results, because the inaccuracies of the measuring devices are now eliminated. Furthermore, it should be noted that all of the valves have been chosen carefully, to minimize dead volumes and adsorption or absorption by seals. All of the tubing used in the setup has been electro-polished, again to minimize adsorption effects. 5. Results Using the new setup, several experiments with methane as carrier gas and water and/or n-nonane have been performed. First, the results of the binary mixtures (methane-n-nonane and methane-water) will be presented. These results will be compared with the different theories previously mentioned1,2,15, as well as with existing experimental data.5 Then, the results of the ternary mixture (methane-water-n-nonane) will be presented. The results will be analyzed by comparing them with the experimental results of the two binary mixtures. All of the experimental data are tabulated in Appendix B. Throughout the next sections the subscripts nm, wm and nwm will be used to indicate mixtures of n-nonane in methane, water in methane, and n-nonane and water in methane, respectively. 5.1. Binary Mixtures. Methane-n-Nonane. In Figure 2, the new nucleation rate data of n-nonane in methane at 235 K and 10 bar are shown, together with different theoretical predictions. The data obtained by Luijten et al.5 which are also shown are obtained at the same pressure (10 bar) but at a different temperature. The new data are consistent with the data by Luijten et al.5 Looking at the new data and the data by Luijten et al., none of the theories shown give good quantitative results, although qualitatively the agreement is reasonable. Because the new data and the data by Luijten et al.5 are at different temperatures, they can be used to check the temperature dependence of the different models. In Figure 3, the ratio of the experimental nucleation rate to the theoretical one is shown for the two different temperatures. It should be noted here that the predictions of the CNT and BCNT are of comparable magnitude, the reason of which we will discus later. It seems that for these conditions the temperature dependence is best predicted by the ICCT model. Another apparent fact is that the scatter in the data at 235 K (new data) is much less than the scatter at 240 K (Luijten et al.5), showing that the new setup has improved the accuracy of the measurements. In Figure 4, the new data of n-nonane in methane are shown together with the data obtained by Luijten et al.5 for a pressure
Peeters et al
Figure 2. Nucleation rate as a function of the supersaturation for n-nonane in methane at 10 bar and 235 K (new data) and 240 K (Luijten). Open squares: Luijten, ref 5. Solid squares: new data. Solid line: SPMT, ref 2. Dashed line: BCNT, ref 1. Long dashed line: ICCT, ref 15. X: new data at 240 K and 10 bar.
Figure 3. Ratio of experimental to theoretical nucleation rate as a function of the temperature for n-nonane in methane at 10 bar. 235 K: new data. 240 K: Luijten, ref 5. X: new data at 240 K and 10 bar.
of 40 bar and 240 K. At these conditions, Luijten’s data show a significant scatter. We checked whether the system was close to the spinodal at these conditions, using the RKS EOS.17 It was found that these conditions, although extreme, were still far from the spinodal. In light of this, and the new experimental results, it is clear that the scatter was caused by uncertainties in the mixture composition. These uncertainties were the highest for the highest pressures, because then the vapor fractions were the smallest. It is also clear from Figure 4 that the theories shown are not able to predict, quantitatively nor qualitatively, the nucleation rates at these conditions, because both the absolute values as well as the slopes of the curves are well off. Methane-Water. In Figure 5, nucleation rates of water in methane vs supersaturation are shown. They are obtained at 10 and 25 bar and 235 K and 40 bar and 240 K. To make a comparison with the ICCT model, several thermodynamic properties of the methane-water system have to be used. One has to realize, however, that the liquid water is at supercooled conditions, for which no experimental data are available. For equilibrium compositions, one can use sophisticated equations of state. The decrease in surface tension due to the presence of
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J. Phys. Chem. B, Vol. 105, No. 47, 2001 11767 TABLE 2: Validation of Conditions (11) and (12) n-C9H20-CH4
10 bar, 235 K 25 bar, 235 K 40 bar, 240 K
Figure 4. Nucleation rate as a function of the supersaturation for n-nonane in methane at 240 K and 40 bar. Open squares: Luijten et al., ref 5. Solid squares: new data. Solid line: SPMT, ref 2. Dashed line: BCNT, ref 1. Long dashed line: ICCT, ref 15.
Figure 5. Nucleation rate as a function of the supersaturation for water in methane. Markers: new data. Lines: ICCT.
TABLE 1: Composition of the Critical Cluster n-C9H20-CH4 10 bar, 235 K 25 bar, 235 K 40 bar, 240 K
H2O-CH4
n/V
n/g
n/V
n/g
21 ( 1
15a
5(2 5(2
4
22b
23 ( 1 16 ( 1 15 ( 1
a Assumed to be equal to the result by Luijten et al.5 at 240 K and 10 bar. b Taken from Luijten et al.5
a high-pressure carrier gas can be accounted for by an adsorption model.7,15 Again, the physical properties are given in Appendix A. Using these properties, we find a quantitative difference of about 5 orders of magnitude and a slight difference in the slope of the theoretical and experimental curves. The latter can be explained by the uncertainty in the surface tension due to the large extrapolation (surface tension fitted to experimental data in the range of 275 to 323 K). Cluster Composition. Using eqs 4 and 5, the number of vapor and gas molecules in the critical cluster can be calculated. These relations have been applied to our new data, and the results are listed in Table 1. The number of methane molecules in the critical cluster of water was obtained by analyzing the results
(2/3)(ngMg)/ (ngMg + nvMv)
|ln(xˆ v/xeq v )|/ (ln Sv)
0.055
0.014
0.27
0.076
H2O-CH4 (2/3)(ngMg)/ |ln(xˆ v/xeq v )|/ (ngMg + nvMv) (ln Sv) 0.11 0.15
0.014 0.050
at a supersaturation Swm ) 10, whereas the equilibrium molar fraction of methane in water was varied between 0.01 < xeq < 0.15, which is incorporated in the given uncertainties. We have compared the experimental results to one-component theories, without having checked if this is at all (physically) justified. Therefore, we will do this now, using the criterion derived before. We shall approximate the ratio of the partial molecular volumes by the ratio of the molecular masses (Mg/Mv ) Vlg/Vlv) to evaluate eq 11. For the evaluation of eq 12, the equilibrium bulk liquid fractions and the fractions in the bulk critical cluster are calculated using the RKS EOS for the methane-n-nonane system. For the methane-water system, an equilibrium fraction of 15% of methane in the bulk liquid is taken, as an upper bound, corresponding to values which can be encountered in methane hydrates. The bulk cluster composition is now approximated using the values given in Table 1. This will render the composition of the cluster as a whole (bulk + surface). At this moment, we have no means of determining the composition of the bulk cluster separately. In all cases, the ratio γˆ /γeq will be taken equal to unity. The variation in ln Sv for each series is relatively small; therefore, a mean value is used. The results are listed in Table 2. From the results listed in Table 2, it can be concluded that the application of a quasi-one-component theory at the conditions of 10 and 25 bar, and 235 K, is reasonable. This also explains why the results of the CNT and BCNT in Figure 3 are comparable. Exact correspondence of CNT and BCNT is not to be expected, because for the CNT the properties listed in the appendix were used, whereas in the BCNT, the liquid density resulting from the solution of the RKS EOS (with Peneloux and Rauzy correction17) and a MacleodSugden correlation17 for the surface tension were used. At 40 bar and 240 K, a one component theory no longer suffices, and the gas-vapor system has to be described by a binary theory. The following question then arises: Why do the BCNT and the SPMT clearly fail at these conditions for the methane-nnonane system? First of all, it is questionable if a macroscopic theory is at all applicable when a critical cluster consists of only four vapor molecules, together with 22 gas molecules. The large difference in the theoretical and experimental slope indicates a breakdown of surface tension at these conditions, which is clearly not predicted by the theory. Furthermore, in the theory as it is applied here, the fraction of gas molecules in the clusters xˆ g is calculated taking the chemical potential of the liquid phase at the internal pressure of the cluster. The result is the fraction of gas molecules of the bulk liquid of the cluster, without a correction for the surface layer of the cluster. As a result, the theoretical gas fraction in the cluster is always smaller than the equilibrium bulk liquid gas fraction,15 whereas the measurements clearly show that the fraction of gas molecules in the cluster (bulk liquid + surface) can be much larger than the equilibrium bulk liquid gas fraction. To present measured binary nucleation data in a condensed form and to provide a basis for the analysis of the ternary nucleation rates, curves were fitted through the results of the binary systems. As a basis for the fit function, the ICCT model (eq 14) was used. In this model, K and θ are only functions of the total pressure and temperature. Hence, for each investigated
11768 J. Phys. Chem. B, Vol. 105, No. 47, 2001
Peeters et al
TABLE 3: Fitted and Theoretical Parameters of ICCT n-nonane in methane θ fit
theory
water in methane θ
log K fit
theory
fit
theory
log K fit
theory
10 bar, 235 K 15.65 15.60 23.72 26.31 10.50 10.81 22.35 27.38 25 bar, 235 K 8.34 9.67 18.75 27.47 40 bar, 240 K 3.76 10.25 15.41 28.23 7.35 9.30 19.00 27.99
Figure 7. Ratio of experimental to fitted nucleation rate of water in methane at 10 bar and 235 K, as a function the supersaturation.
Figure 6. Ratio of experimental to fitted nucleation rate of n-nonane in methane at 10 bar and 235 K, as a function the supersaturation.
(binary) series, they should be constant, and they are now used as fit parameters. As seen in the previous section, this is physically meaningful for the 10 bar measurements (and 25 bar measurements). However, for the 40 bar measurements, this should only be regarded as a convenient fit function, without any real physical significance. The results are listed in Table 3. For comparison, the values of K and θ, using the thermodynamic properties given in Appendix A, are also listed. It is clear from Table 3 that the discrepancies between theory and experiment can be quite large. However, one should bear in mind that this does not necessarily mean that the nucleation theory is wrong, but discrepancies can also originate from inaccurately known thermodynamic properties. To give an idea of the accuracy of the fit through the data, the ratio of the experimental nucleation rate to the fitted one is shown as a function of the supersaturation for each of the binary series, see Figures 6-9. From these figures, it can clearly be seen that the scatter around the fit function remains within a factor 2 in terms of nucleation rates, for all series. 5.2. Ternary Mixtures. The experiments with the ternary mixture methane-n-nonane-water have been performed at two different nucleation conditions, those being 10 bar and 235 K and 40 bar and 240 K. The small difference in temperature was necessary in order to investigate the full range of possible compositions at both pressures. The goal was to investigate whether the presence of one supersaturated vapor component would influence the nucleation rate of the other. In other words, to see if any conucleation would take place. This was investigated by keeping one of the vapor fractions constant while increasing the fraction of the other vapor component step by step. The nucleation rate of the ternary mixture was then compared to the sum of the nucleation rates of the two binary systems at the same conditions. What will be plotted is the ratio of the experimentally found ternary nucleation rate to the sum of the (fitted) binary nucleation rates. This is shown in Figures 10-13. In all cases, the scaled ternary nucleation rates are within
Figure 8. Ratio of experimental to fitted nucleation rate of n-nonane in methane at 40 bar and 240 K, as a function the supersaturation.
Figure 9. Ratio of experimental to fitted nucleation rate of water in methane at 40 bar and 240 K, as a function the supersaturation.
a factor 4 of the ideally noninteracting value of 1. Therefore, one can conclude that in the mixture methane-n-nonane-water the two vapors do not (or hardly do) influence each other in the nucleation process. Considering the macroscopic properties
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J. Phys. Chem. B, Vol. 105, No. 47, 2001 11769
Figure 10. Ratio of experimental ternary nucleation rate to the sum of the fitted binary nucleation rates, at 10 bar and 235 K, as a function the n-nonane supersaturation.
Figure 12. Ratio of experimental ternary nucleation rate to the sum of the fitted binary nucleation rates, at 40 bar and 240 K, as a function the n-nonane supersaturation.
Figure 11. Ratio of experimental ternary nucleation rate to the sum of the fitted binary nucleation rates, at 10 bar and 235 K, as a function the water supersaturation.
Figure 13. Ratio of experimental ternary nucleation rate to the sum of the fitted binary nucleation rates, at 40 bar and 240 K, as a function the water supersaturation.
of n-nonane and water, this might not seem very surprising, because these are two highly immiscible components. However, one should bear in mind that in the cases studied both vapor components were in a (highly) supersaturated state and that the clusters are not of macroscopic but microscopic sizes. Nucleation rates of mixtures of n-nonane and water have been studied before by Wagner and Strey18 and Viisanen and Strey.19 They measured the nucleation rates in combination with a inert carrier gas, at 230 and 240 K and atmospheric pressures. They also found that the two vapors nucleate independently. Furthermore, Ten Wolde and Frenkel performed a numerical (MC) study on the nucleation behavior of poorly miscible systems.20 They found that the critical clusters formed in supersaturated vapors of highly immiscible liquids were always rich in one of the two components. This indicates independent nucleation of the components, which is again in agreement with the experimental results.
n-nonane in methane at 40 bar and 240 K have shown that the scatter previously found in measurements at these conditions was mainly due to inaccurately known vapor fractions. From these new measurements at 40 bar and 240 K, it is now even more clear that the interpretation of these results require new and improved models, because existing models clearly fail at these extreme conditions. The measurements of nucleation rates of water in methane have not been measured before (to our knowledge). From the new binary data, the composition of critical clusters could be determined, using the nucleation theorem. These cluster compositions were then used to check whether the binary gas-vapor systems could be described by a one component theory. For the data at 10 and 25 bar, this indeed is the case. The remaining discrepancies between the one component theory and the experiments indicate that the theory is incomplete (n-nonane in methane) or that the physical properties are incorrect (water in methane). Furthermore, experimental error cannot be completely excluded, because no independent data are available for comparison. At 40 bar, a one component theory is no longer applicable for the systems investigated here, and a binary theory has to be used. It is essential that this binary theory correctly incorporates the surface region/molecules of the cluster. After analyzing the nucleation
6. Conclusions With the newly developed mixture preparation device accurate high-pressure nucleation data for n-nonane and/or water in methane have been obtained. The new measurements of
11770 J. Phys. Chem. B, Vol. 105, No. 47, 2001
Peeters et al
rates of the ternary mixture methane-n-nonane-water, it can be concluded that the two vapor components nucleate independently. This is in agreement with existing measurements at low pressure and numerical studies of nucleation in mixtures of poorly miscible supersaturated vapors.
Water. Molar mass:17
7. Appendix A: Physical Properties
Pure component saturated vapor pressure:23
M ) 0.018 015 kg mol-1
When the ICCT model is expressed as
(
J ) KS exp θ -
[
)
3
ps ) 610.8 exp -5.1421 ln
4 θ 27 (ln S)2
(15)
T (273.15 )1 1 6828.77( T 273.15)]
(Pa)
the parameters K is
K)
Liquid density:24
( )( ) s
2
p fe ZgkBT
2σ M πNA
1/2
1 Fl
(16)
Fl ) 999.84 + 0.086(T - 273.15) - 0.0108(T - 273.15)2 (kg m-3)
and the parameter θ is
( )
θ ) (36π)1/3
M NAFl
2/3
Surface tension:15,25-28
σ kBFl
(17)
( )
σ ) σ0 - nakBT ln
Here, Zg, kB, and NA are the compressibility of the methane, Boltzmann’s constant, and Avogadro’s number, respectively. The values of the other parameters depend on the vapor (and carrier gas) that is being used. n-Nonane. Molar mass:17
M ) 0.128 259 kg mol
9467.4 + 128.778 89 T
pL ) -481.95 + 2.1211T
(bar)
)
(Pa)
Fl ) 733.503-0.787 562(T - 273.15) - 9.689 37 × 10-5 (kg m-3)
(T - 273.15)2 tension:22
(N m-1)
na ) 6 × 1018 m-2 pL ) 258.5 - 2.731T + 7.781 × 10-3T2
(bar)
Enhancement factor:22
fe ) exp[b(p - ps) + c(p - ps)2]
(bar-1)
c ) -1.2070 × 10-4 + 2.6726 × 10-2/T + 2.3585/T2 (bar-2)
8. Appendix B: Experimental Data Tables 4-6 show data of n-nonane in methane, water in methane, and water and n-nonane in methane. TABLE 4: Data of n-Nonane in Methane
(N m-1)
σ0 ) 0.02472-9.347 × 10-5(T - 273.15)
(-)
b ) -7.0874 × 10-3 + 3.4131/T + 1.400/T2
Liquid density:21
p + pL pL
(N m-1)
na ) 5.4 × 1018 m-2
fe ) exp[b(p - ps) + c(p - ps)2]
(
( )
σ0 ) 0.127 245 - 1.898 45 × 10-4T
Enhancement factor:29
ps ) 133.322 exp -17.568 32 lnT + 0.015 255 6T -
σ ) σ0 - nakBT ln
(N m-1)
-1
Pure component saturated vapor pressure:21
Surface
p + pL pL
(-)
b ) 2.385 × 10-1 - 7.26 × 10-4T - 1.29 × 10-6T2 + 4.61 × 10-9T3 (bar-1) c ) 2.424 × 10-2 - 2.552 × 10-4T + 8.985 × 10-7T2 1.057 × 10-9T3 (bar-2)
p yw × 105 yn × 105 (bar)
T (K)
J (m-3s-1)
10.02 9.98 10.02 10.15 10.02 9.96 10.03 9.99 9.95 10.06
234.57 235.13 234.87 235.32 235.48 234.57 234.81 235.36 234.70 235.10
6.0 × 1015 5.5 × 1015 2.0 × 1016 2.9 × 1016 4.9 × 1016 2.5 × 1015 1.0 × 1015 2.3 × 1014 3.0 × 1014 8.8 × 1013
11.34 7.75 6.33 4.92 9.26
40.05 39.95 39.94 40.04 39.95
240.38 240.28 240.34 240.24 240.53
2.8 × 1017 4.0 × 1016 1.3 × 1016 5.3 × 1015 7.0 × 1016
37.41
10.17 240.69 1.0 × 1017
run
p0 (bar)
T0 (K)
302 303 304 305 306 307 308 309 310 311
24.71 24.71 24.73 24.74 24.73 24.70 24.71 24.70 24.71 24.73
293.75 294.65 294.15 293.85 294.85 294.15 293.95 294.85 294.35 294.15
24.14 25.05 25.99 27.04 28.15 23.31 22.52 21.81 21.11 20.47
340 341 342 343 344
88.13 88.43 88.43 88.43 88.43
296.05 296.35 296.45 296.15 296.65
028 23.05 294.85
High Pressure Nucleation Experiments
J. Phys. Chem. B, Vol. 105, No. 47, 2001 11771 References and Notes
TABLE 5: Data of Water in Methane run
p0 (bar)
T0 (K)
p (bar)
T (K)
yw × 105
(m-3 s-1)
103 104 105 106 107 108 170 195 201
24.57 24.60 24.57 24.58 24.50 24.55 24.92 24.50 24.65
293.85 294.25 294.55 294.45 294.75 294.55 295.45 293.95 295.25
24.11 28.93 33.10 33.04 31.09 26.61 32.90 32.45 30.20
9.95 9.95 10.01 9.98 10.02 9.97 10.06 9.85 10.01
234.57 234.86 235.54 235.28 235.98 235.32 235.76 234.28 235.92
1.6 × 1014 4.6 × 1015 4.5 × 1016 5.0 × 1016 2.2 × 1015 1.7 × 1014 2.4 × 1016 1.7 × 1017 1.1 × 1015
114 115 116 117 118 119 120
60.02 60.07 60.09 60.12 60.06 60.05 60.10
294.75 295.15 294.85 295.15 294.75 294.85 295.05
12.07 12.04 11.51 10.29 9.04 9.67 10.88
24.89 25.22 25.09 25.07 25.33 25.03 25.06
234.49 235.59 235.00 235.15 235.52 234.88 235.07
1.5 × 1017 3.3 × 1016 2.0 × 1016 4.6 × 1015 2.1 × 1014 2.0 × 1015 8.6 × 1015
173 174 185 186 187 188 189 190 191 192
87.39 87.29 87.40 87.72 87.80 87.89 87.91 86.97 87.53 87.99
295.45 294.75 295.15 295.55 295.95 295.85 296.15 294.55 295.35 295.55
9.41 10.21 9.09 9.61 8.98 9.86 10.26 10.24 10.55 10.16
40.13 40.00 39.94 40.17 39.93 39.75 40.06 40.07 40.21 40.10
240.55 239.81 239.99 240.45 240.35 239.91 240.66 239.99 240.48 240.13
1.3 × 1016 1.3 × 1017 1.3 × 1016 2.5 × 1016 7.7 × 1015 4.2 × 1016 2.5 × 1016 1.4 × 1017 1.1 × 1017 8.0 × 1016
yn × 105
J
TABLE 6: Data of Water and n-Nonane in Methane run
p0 (bar)
T0 (K)
yw × 105
yn × 105
p (bar)
T (K)
J (m-3 s-1)
253 252 251 203 204 205 206 208 209
24.76 24.56 24.50 24.67 24.70 24.71 24.69 24.72 24.72
294.45 294.45 294.25 295.25 295.15 295.25 295.35 294.55 294.05
31.54 31.40 31.41 30.91 31.05 31.17 30.53 30.49 30.36
0.97 1.94 2.92 4.99 7.56 10.12 12.97 18.19 20.78
10.02 9.95 9.86 9.97 9.94 10.00 10.00 10.15 9.99
235.03 235.13 234.60 235.63 235.34 235.74 235.86 235.98 234.60
4.0 × 1016 2.2 × 1016 4.3 × 1016 7.5 × 1015 2.0 × 1016 1.0 × 1016 5.3 × 1015 7.0 × 1015 6.1 × 1016
225 224 223 222 221 220
24.73 24.73 24.74 24.74 24.76 24.76
294.55 294.75 295.25 295.35 295.35 295.15
29.63 24.07 19.29 15.45 9.53 4.77
22.96 23.35 23.36 23.02 23.35 23.29
10.00 9.99 10.00 10.01 9.95 10.10
235.08 235.17 235.69 235.82 235.41 236.10
6.6 × 1015 1.7 × 1015 6.7 × 1014 4.4 × 1014 1.0 × 1015 4.3 × 1014
230 231 232 234 240 241 242
88.70 88.80 87.59 87.80 88.10 87.99 88.02
296.65 296.65 295.25 295.35 295.65 295.55 295.95
1.86 3.79 5.86 9.83 9.81 9.61 9.61
8.25 8.03 7.90 7.97 6.29 4.71 3.18
40.03 39.95 39.85 39.97 39.89 40.14 40.24
240.45 240.25 239.78 239.89 239.81 240.20 240.69
6.3 × 1016 5.8 × 1016 2.5 × 1016 6.6 × 1016 7.3 × 1016 3.8 × 1016 1.2 × 1016
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