ISSN 1063-7842, Technical Physics, 2006, Vol. 51, No. 4, pp. 423–430. © Pleiades Publishing, Inc., 2006. Original Russian Text © A.A. Brin, S.P. Fisenko, 2006, published in Zhurnal Tekhnicheskoœ Fiziki, 2006, Vol. 76, No. 4, pp. 31–38.
THEORETICAL AND MATHEMATICAL PHYSICS
Simulation of the Operation of a Laminar-Flow Diffusion Chamber to Study Homogeneous Nucleation: Part II A. A. Brin and S. P. Fisenko Luikov Institute of Heat and Mass Transfer, National Academy of Sciences, Minsk, 220728 Belarus e-mail:
[email protected] Received April 20, 2005
Abstract—A new mathematical model of the operation of a laminar-flow diffusion chamber is studied numerically. The domains of applicability of the standard mathematical model, which have been qualitatively estimated earlier, are confirmed. The nature of the carrier gas is found to substantially influence the volume of the nucleation zone in the chamber. In particular, the nucleation zone volume in the case of argon as a carrier gas is one order of magnitude larger than that in the case of helium, all other things being equal. Analysis of Vohra and Heist’s experiments using the new mathematical model shows that nonlinear interaction between growing droplets is of key importance at a relatively high vapor supersaturation. The role of a locally nonuniform supersaturation field arising near droplets in the nucleation zone is estimated. PACS numbers: 02.70.-c, 45.15.-x DOI: 10.1134/S1063784206040050
INTRODUCTION
expressed as
A new mathematical model of the operation of a laminar-flow diffusion chamber (LFDC) was presented in [1]. In this paper, we report the results of numerical simulation in terms of this model and compare them with the experimental data. The effects related to the formation of a locally nonuniform field of the supersaturated vapor near droplets in the nucleation zone are discussed on a semiquantitative basis.
Tw u f = u 0 ------. T in
For the transition stage (i.e., in the course of cooling the mixture), we employ the simplest, linear approximation of the average velocity. Indeed, temperature differences in the LFDC are relatively small. Neglecting slight changes in the velocity profile [2] and taking into account only a change in the flow average velocity, we obtain ⎧ ( u f – u 0 )z ⎪ u 0 + ----------------------- , L u(z) = ⎨ ⎪ u , z ≥ L, ⎩ f
1. PROCEDURE AND RESULTS OF NUMERICAL SIMULATION Before presenting numerical results, we will approximately calculate the velocity of the vapor–gas mixture, which changes when the mixture cools down in the LFDC condenser. Calculation is based on the integral continuity equation for the vapor–gas mixture, ρ ( T in )u 0 = ρ ( T w )u f ,
(1)
where Tw is the temperature of the condenser; Tin is the temperature of the mixture at the chamber inlet; ρ(Tw) and ρ(Tin) are the densities of the mixture at temperatures Tw and Tin, respectively, under atmospheric pressure; and uf is the final velocity of the mixture. We will also use the equation of state of ideal gas. After cooling, final velocity uf of the mixture in the chamber is
(2)
z
(3)
where z is the coordinate and L is a fitting parameter (its value is found by iteration). The iterative process is assumed to converge if the temperature of the vapor– gas mixture drops by value Tw over length L along the chamber axis. We retain the assumption that the flow velocity profile is parabolic and so only the average velocity changes [3]. To solve the set of partial and ordinary differential equations, we use the method of semidiscrete straight lines [2, 4]. In the framework of this method, we use expansion in radial variable to obtain finite differences with allowance for variable transport coefficients; independent variable z is taken to be continuous. As a result, a set of partial differential equations is approximated by
423
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424 Dimensionless density, n/ns 5
4
1
1.08 1
2
3
3
2 1
Dimensionless temperature, T/Tω 1.12
2
1.04
3
0
1
2 3 4 5 Dimensionless axial position, z/R
Fig. 1. Propanol dimensionless density profiles (1) along the chamber axis, for r/R = (2) 0.286, and (3) 0.714.
a set of ordinary differential equations (ODEs). Since this procedure is applied to any of the two coupled equations that describe the temperature and vapor density fields, the number of ODEs is equal to double the number of nodes in the radial direction plus additional equations describing growth. It is well-known from the kinetic theory of gases that the transport coefficients nonlinearly depend on the composition and temperature of the mixture [5]. Thus, we have a nonlinear set of ODEs. The set of ODEs thus obtained was solved numerically using the MathCAD 2000 Professional software package and the fourth-order Runge–Kutta method. Figures 1–4 show the data obtained when a set of 14 ODEs was used to describe transport processes in the gas phase. The thermal physical properties of propanol and carrier gases, as well as the thermal conductivity and the diffusion coefficient of propanol in helium and other gases, were taken from [6, 7]. The thermal conductivity of a binary mixture was calculated using the methods described in [8]. Figure 1 shows the profiles of the propanol vapor density (the carrier gas is helium). These profiles were calculated in terms of the standard mathematical model, which ignores the effect of growth on the state of the vapor–gas mixture. To make the vapor density dimensionless, it was related to the propanol saturated vapor density at temperature Tw, which was used as a density scale. As a spatial scale, we took the condenser radius. Note that, early in the chamber axis, the vapor density is constant. As is seen from Fig. 1, the vapor density decreases exponentially, in agreement with the qualitative estimates made above. Figure 2 shows the dimensionless temperature profiles of the vapor–gas mixture along the chamber at var-
1.00
0
0.25 0.50 0.75 Dimensionless axial position, z/R
Fig. 2. Dimensionless temperature profiles for the propanol–helium mixture (1) along the chamber axis, for r/R = (2) 0.286, and (3) 0.714.
ious distances from the axis (the carrier gas is helium). As the temperature scale, we used temperature Tw. The temperature is seen to decrease exponentially even at z > 0.12R. The run of curves 3 in Figs. 1 and 2 suggests that local zones of small supersaturation appear near the entrance of the condenser away from the central part of the flow because of a rapid decrease in the mixture temperature away from the center of the channel. Local maxima of the supersaturation in these zones are caused by the exponential temperature dependence of the saturated vapor density. This fact was first discovered in [6]. Figure 3 shows the growth of clusters formed near the chamber axis for different mixtures with the same temperature and average velocity. For the helium-containing mixture, a droplet has radius of 5 µm at a distance of 4R from the condenser inlet. For the argon– propanol mixture, the nucleation rate is maximum at a distance of 6R, whereas a droplet takes a radius of 2 µm at a distance of 10R. Note that growth in the free molecular regime (Kn 1) is so fast that particles virtually stay at their positions. The diffusion growth of relatively large droplets occurs at a substantially lower rate throughout the chamber. The results of numerical simulation are in good qualitative agreement with the analytical estimates made in [1]. Figure 4 plots the global minimum of dimensionless free energy ∆Φ* of critical propanol clusters against temperature difference Ts – Tw. Only Tw was varied in the calculations; the other temperatures remained constant (Tin = 330 K, Ts = 322.5 K). By the global minimum of free energy ∆Φ*, we mean the lowest free energy of critical cluster formation calculated in terms of the capillary approximation [9] in the nucleation zone. As follows from Fig. 4, critical cluster nucleation TECHNICAL PHYSICS
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Global minimum of free energy, ∆Φ* 250
Droplet radius Rd, 10–6 m 1
200
5
150 3 2
100 50
1 0
4 8 12 Dimensionless axial position, z/R
0 15
20
25 30 35 Temperature difference, K
Fig. 3. Growth of droplets in the LFDC: (1) helium–propanol mixture and (2) argon–propanol mixture.
Fig. 4. Global minimum ∆Φ* of the free energy of critical cluster formation vs. temperature difference Ts – Tw.
rate Jc is negligibly small when the temperature difference is less than 25 K, since Jc ~ exp[–∆Φ*]. As the temperature difference increases, the free energy of critical cluster formation decreases substantially primarily because of an increase in the vapor supersaturation with temperature difference. From the classical theory of nucleation kinetics, one may expect that the nucleation rate in the temperature difference range considered in Fig. 4 will increase by about 20 orders of magnitude. Note that the calculations shown in Fig. 4 employ the parameters of the first experiment (the temperature and mixture flow rate) conducted in [6]. For the same experimental conditions, Fig. 5 shows the profiles of dimensionless free energy ∆Φ(r, z) of critical propanol clusters in the nucleation zone. It is seen that the value of ∆Φ* differs from the qualitative estimates made in [1, Fig. 3] only slightly. As follows from Fig. 5, the nucleation zone in the radial direction does not extend beyond 0.2R. The numerical results given in Figs. 1–5 and Tables 1 and 2 were obtained without regard for the depletion and heating of the vapor–gas mixture by growing droplets. It is obvious that these effects will substantially influence the density and temperature fields at a sufficiently high concentration of droplets. To jointly simulate nucleation and heat-and-mass transfer in the LFDC, we place fine droplets (Rd ~ 2 × 10–9 m), which are able to grow, at the condenser’s cross section having fixed coordinate z. In our statement of the problem, the number of such droplets is a free parameter, which makes our model applicable to analysis of flows with heterogeneous nucleation as well. Basically, the number of droplets can be calculated using nucleation theories. In the first iteration, we introduce a group of droplets and calculate both their growth and their influence on the gas phase parameters. Using the results obtained, we determine a new position
of ∆Φ*. In the second iteration, we place the same group of droplets on the same position, then place the second group of droplets in downstream direction but in front of the new global minimum of the free energy, calculate the growth of the droplets, and analyze the gas phase. In this way, we simulate the effect of an integral source in terms of our mathematical model [1].
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Figure 6 plots ∆Φ* versus the total number of droplets having formed in the chamber. At the place the droplets were introduced, the free energy differs from global minimum ∆Φ* obtained in the previous iteration (in our case, in the framework of the standard model) by two units. It is seen that, if the density of clusters having formed in front of the global minimum is high, growing clusters disturb the vapor–gas mixture. As a Free energy of critical cluster formation, ∆Φ 30 28 3 26 24
2
22
1 0
1 2 Dimensionless axial position, z/R
Fig. 5. Profiles of the dimensionless free energy of critical cluster formation in the nucleation zone (1) along the chamber axis, for r/R = (2) 0.286, and (3) 0.429.
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426 ∆Φ* 70
Droplet radius Rd, 10–6 m 1.2
60
1.0
2
0.8
4 1 3
2
50
0.6 40
0.4 0.2
30 1 20
106
107
108
109
0 0.25
1010
0.30
Number of clusters, Nd Fig. 6. Global minimum ∆Φ* vs. the number of clusters at the beginning of the nucleation zone: (1) helium–propanol mixture and (2) argon–propanol mixture.
0.35 0.40 0.45 0.50 Dimensionless axial position, z/R
Fig. 7. Cluster growth in the LFDC: (1, 3) the first group of droplets (Nd = 108 clusters/(m3 s)) for radial positions (r/R = 0.143 and 0.284) and (2, 4) the second group of droplets (N2 = 1010 clusters/(m3 s)) for the same radial positions.
result, ∆Φ* increases, causing the nucleation rate to drop under the same condition of the vapor–gas mixture at the condenser. For the conditions of the first experiment in [6], our calculations demonstrate that the standard mathematical models of LFDC works well when droplet density Nd is less than 1018 m–3. This value agrees with our estimate of Nd [1, Eq. (13)]. Interestingly, an average nucleation rate as low as about 1010 clusters/(s m3) suffices to reach a cluster density of 108 m–3 within 0.1R from the inlet. This effect is more pronounced in the argon–propanol mixture (Fig. 6, curve 2), since it has a lower thermal diffusivity.
from the curves, the droplets moving near the chamber axis first grow and then start evaporating, whereas those moving away from the chamber axis, where heat exchange with the wall is favored, continue growing. Such behavior is accompanied by a substantial increase in ∆Φ*. Thus, application of the standard mathematical model to highly nonequilibrium systems in the LFDC causes significant errors. However, we will use the standard mathematical model of LFDC in the next section to analyze the experiments conducted in [6], since the nucleation rates detected are small.
Below, we will present the calculation results obtained when the integral terms in the equations of convective diffusion and heat conduction are “split” into two groups of droplets. The finer the discretization, the higher the accuracy of simulation of homogeneous nucleation in the LFDC. The results of such a calculation are illustrated in Fig. 7, which plots the radius of droplets of either group versus their position in the chamber. It is seen that the heat of phase transition decreases the supersaturation and slows down the growth. Significantly, the temperature at the site the droplets enter the system is well above Tw. As follows
2. CALCULATION RESULTS VERSUS EXPERIMENTAL DATA Table 1 lists the results of numerical simulation of the nucleation zone parameters for the experiments with the helium–propanol mixture [6]. The parenthetical figures in the first column indicate the number of the experiment performed from [6]. At the given temperatures and flow rates, the degree of supersaturation is extremely high. Recall that the nucleation rate was about 106 particles/(m3 s) for all the supersaturations found experimentally. The calculated and experimental data are seen to be in conflict with each other. Note that
Table 1. Helium–propanol mixture for the experiments in [2]
Table 2. Argon–propanol mixture for the experiments in [2]
N
∆Φmin
Z
D
S
Tmin
N
∆Φmin
Z
D
S
Tmin
1(1) 2(5) 3(17) 4(22) 5(27)
21.8 23.6 21.2 16.4 15.0
0.9 0.8 1.1 1.3 2.1
1 1 1 1.1 1.5
4.4 4.3 4.1 5.0 4.9
295.4 292.9 301.7 302.4 307.4
1(28) 2(32) 3(37) 4(40) 5(44)
48.4 43.8 49.6 55.2 48.3
7.4 8.0 8.3 7.6 8.0
2.5 2.7 2.7 2.4 2.7
2.4 2.5 2.4 2.3 2.4
309.5 310.8 308.9 306.9 308.7
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the width of the nucleation zone calculated for the helium–propanol mixture in terms of the standard mathematical model roughly equals R. The second column gives the value of global minimum ∆Φ*; the third column, the dimensionless position of ∆Φ* in the condenser; the fourth column, the dimensionless width of the nucleation zone; the fifth column, the maximum supersaturation; and the sixth column, the temperature of the mixture at the position of global minimum ∆Φ*. To obtain the dimensionless quantities, we divided position Z and width D of global minimum ∆Φ* by chamber radius R. For the argon–propanol system, Table 2 presents the nucleation zone parameters calculated from the experimental data of Table 2 given in [6]. The columns of Table 2 list the same quantities as those of Table 1. The calculations were carried out with a program developed by us in terms of the standard mathematical model of LFDC. It is seen the properties of the carrier gas substantially affect the temperature and vapor density fields and, hence, the value of ∆Φ* through the Lewis number Le. In the case of argon as a carrier gas, the nucleation zone size is substantially larger. Comparing our results with the calculation and visualization performed in [6] demonstrates that the temperature fields in the LFDC condenser found in [6] are incorrect. The temperature is overestimated in [6]; specifically, the error is about 5°C at the channel axis. This discrepancy is likely to be caused by the fact that the numerical calculations did not take into account the variation of the transport coefficients and also by the fact that the authors of [6] chose an improper scheme for numerical calculation. At a constant flow rate of the mixture, the particle radius depends on the LFDC saturator temperature (experiments 1, 5, 17): as the temperature increases, the radius increases, which agrees with our conclusions [1]. Our numerical simulation indicates that the nucleation zone radius for argon is 0.4R, where R if the LFDC condenser radius. If helium is used as a carrier gas, the nucleation zone radius is ≈0.25R. Thus, for the experiments with argon, the nucleation zone volume is approximately four times that for the experiments with helium. If “feedback” between the droplets that grow and move in the nucleation zone and the global minimum of the free energy is taken into account, the nucleation zone volume decreases substantially. This decrease, however, is much greater for the experiments with helium. From Tables 1 and 2, the calculated nucleation rates in helium and argon are bound to differ by more than ten orders of magnitude. However, the experiment gives approximately the same nucleation rates! We think that the arising (and not taken into account) local structure of the supersaturation field makes comparison between the calculated and experimental data on homogeneous nucleation in the LFDC are totally incorrect. TECHNICAL PHYSICS
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The effect of these structures is considered in the next section. 3. LOCAL STRUCTURE OF THE SUPERSATURATION FIELD Along with the effects involved in our model (a decrease in the vapor content due to condensation on forming droplets, release of the heat of phase transition, and a finite size of the nucleation zone), homogeneous nucleation in the LFDC seems to include other kinetic effects. The first one is related to the microstructure of the supersaturation field near a growing droplet. The authors of [11] were the first to demonstrate the importance of this effect in analysis of experiments with high nucleation rates. Indeed, supersaturation S on the surface of a droplet growing in a vapor–gas mixture is close to unity. For the diffusion growth of droplets, one can easily show that the supersaturation becomes equal to the average supersaturation calculated by our LFDC model only at a distance of about ten droplet radii. Let this distance be Reff. Then, near a growing droplet (at a distance less than Reff), the vapor supersaturation is substantially lower than that calculated by our model. In other words, there exists a local structure in the supersaturation field. Unfortunately, taking into account this effect in our mathematical model makes the problem unsolvable. Therefore, we will make only semiquantitative estimates. It is obvious that a relatively large droplet having formed at the beginning of the nucleation zone moves through this zone with the velocity of 2 the gas flow. In this case, a cylinder of volume vfπ R eff contains one growing droplet in a unit time (for simplicity, Reff is taken to be constant). According to the classical concept of nucleation kinetics [11], Jcvfπ R eff particles (where Jc is the nucleation rate) are expected to form for this time. At low nucleation rates, this effect of local supersaturation field is weak. Indeed, if 2
J c πR eff v f 1, 2
(4)
this effect may be neglected. At high nucleation rates, this effect is significant. From the data in Fig. 3, it follows that Reff ~ 20 µm for the helium–propanol mixture and vf ~ 5 × 10–2 m2/s. Then, at a moderate nucleation rate in the LFDC (Jc ~ 1012 clusters/(m3 s), only one cluster, rather than 60 clusters, is detected in the region of size Reff. Thus, for this example, the discrepancy between the “classical” and experimental nucleation rates is almost two orders of magnitude because of the local structure of the supersaturation field. Obviously, the local-field effect is much stronger in the experiments with helium. In helium mixtures, droplets grow faster than in argon mixtures, because the diffusion coefficient of helium is higher. Thus, the observed nucleation rate in helium is lower than in argon, all
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428
other things being equal. This fact, which was experimentally discovered in Refs. [2–5] cited in [1], is also responsible for the nonlinear dependence of the number of clusters formed in the LFDC on the mixture flow rate. Another effect of local supersaturation field is observed at high degrees of vapor supersaturation (hence, at high nucleation rates) [6]. This effect arises on much shorter scales, on the order of the mean free path of vapor molecules. It is related to the fact that, when the density of new-phase clusters is high, the growth of a cluster cannot be considered as being independent of the growth of its neighbors. Let us make some qualitative estimates. If the cluster average spacing in the condensed phase is d, we have d ∼ 1/N d . 1/3
If d is on the order of mean free path λv of vapor molecules, ( 1/N d )
1/3
∼ λv .
(5)
For Eq. (5) to hold, nucleation rate J must meet the condition J ∼ N d /τ.
(6)
Characteristic nucleation time τ is about 10–6 s [12], and λv ~ 10–6 m for propanol molecules at atmospheric pressure. Then, from Eq. (6), we can estimate the nucleation rate at which Eq. (5) is valid (J ~ 1023 clusters/(m3 s)). Due to the stochastic character of nucleation, clusters appear with a certain time delay and are nonuniformly distributed in the space. Let one of the clusters grow faster than its neighbors. This cluster will suppress the growth of the neighbors for several reasons, such as coalescence [11, 13, 14] or release of the latent heat of phase transition. Large clusters are heated to a less extent because of their higher heat capacity and lower evaporation losses. The numerical estimates based on the classical theory of nucleation kinetics show that a supersaturation at which such high nucleation rates can be observed was reached in the experiments with helium and hydrogen [6]. For the second kinetic effect (which is related to the local supersaturation field), the maximum density of droplets grown in the LFDC condenser, Neff, is estimated to be N eff ≈ 0.25R /R eff . 2
2
The maximum radial size of the nucleation zone is assumed to be half the condenser radius. At the diffusion growth (Kn ~ 0.1), the typical radius of a propanol cluster is ~2 µm (for helium as a carrier gas). For the experiments in [6], we then have Neff ≈ 4 × 104 clusters. The nucleation rate calculated by formula (4) in [1] for this cluster density (i.e., for Neff ≈ 4 × 104) is approximately equal to 6 × 106 clusters/(m3 s), which agrees with the experimental data in [6].
The extrapolation of the experimental data for propanol nucleation rates that have recently been obtained by the nonstationary method [15] supports our viewpoint that an extremely high nucleation rate (1022 clusters/(m3 s), which is much higher than 106 clusters/(m3 s)) was indeed achieved in the experiments [6]. Moreover, in [15], no difference in the experimental data for the nucleation rate in helium and argon was found. It should be emphasized that diffusion interaction between growing clusters [11, 14], which eliminates the majority of fine clusters, restores the density and temperature profiles in the LFDC and, thereby, extends the domain of applicability of the mathematical model developed in [1] for calculating the density and temperature profiles and particle radii. Note that the mean free path of propanol molecules in helium is several times longer than in argon [10]. Therefore, when argon is used as a carrier gas, the second effect seems unlikely. 4. DISCUSSION OF RESULTS We constructed an iterative algorithm to seek for a self-consistent numerical solution for flows undergoing homogeneous condensation in a laminar-flow diffusion chamber. In the course of numerical experiments, we found a zone with a maximal supersaturation that arises near the condenser inlet and partially overlaps with a zone of the minimal free energy of critical cluster formation. Near this minimum, a nucleation zone is located. Its volume was found to substantially depend on the thermal properties of a carrier gas and on the mixture flow rate. In particular, the volume of the nucleation zone in the case of argon as a carrier gas is significantly larger than in the case of helium, all other LFDC parameters being equal. At the same nucleation rate, the number of forming particles is directly proportional to the nucleation zone volume. Our calculations show that this volume increases linearly with a small increase in the flow rate of the vapor–gas mixture. Thus, the paradoxical result on the effect of a carrier gas (see Refs. [2–5] cited in [1]) has been explained in simple physical terms. It was shown that the LFDC can be applied to study the homogeneous nucleation of vapor only at relatively small nucleation rates. In this case, the state of the vapor–gas mixture is not disturbed by growing clusters and the inverse problem of finding a dependence of the nucleation rate on the temperature and supersaturation can be solved correctly. The condition that the volume density of droplets (clusters), Nd, must satisfy the inequality N d Le ( U/kT w ) R R d –1
–2
–1
was obtained analytically and supported numerically. If this condition fails, the growth of droplets formed at the beginning of the nucleation zone (in the downstream direction) causes a significant rise in the temperature at the center of the diffusion chamber. This temTECHNICAL PHYSICS
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Droplet radius Rd, 10–6 m 4
Dimensionless temperature, T/Tω 1.12
2
3 1.08
1
1 2 2
1.04
1 3
1.00
0 0
1
2 3 4 5 Dimensionless axial position, z/R
1
2 3 4 5 Dimensionless axial position, z/R
Fig. 8. Dimensionless temperature profiles at high nucleation rates for r/R = (1) 0.286 and (2) 0.714.
Fig. 9. Effect of the radial position on the growth of droplets in the LFDC. r/R = (1) 0.14 and (2) 0.286.
perature rise drastically shrinks the nucleation zone and, additionally, increases global minimum ∆Φ* of the free energy of critical cluster formation. The latter effect is shown in Fig. 6. An increase in the temperature in the nucleation zone leads to a decrease in the total number of droplets having grown to an optically detectable size (about several microns). It is interesting that this effect is more pronounced in argon mixtures because of a lower thermal conductivity of argon. In the standard mathematical model of the LFDC operation, the nucleation rate is maximal at the chamber axis. At high degrees of supersaturation, taking into account finite dimensions of the nucleation zone and rapid growth of droplets in the free molecular regime renders the center of the channel the hottest region. As a result, the supersaturation at the center decreases because of a rise in the temperature, the nucleation rate drops, and the growth of particles slows down. The maximum of the nucleation rate offsets from the axis, and the nucleation zone volume decreases sharply. Thus, the experimental fact that the number of droplets is maximal away from the condenser axis may be an indication of nonlinear effects. Figure 8 shows the evolution of the temperature of the vapor–gas mixture in an LFDC operating at a high nucleation rate. An increase in the temperature slows down the growth of droplets because of a decrease in the supersaturation, which, in turn, suppresses the liberation of the latent heat of phase transition. As a result, the mixture begins to cool down again but at a rate much lower than at the beginning of the condenser. Naturally, this is because the particles continue to grow and, accordingly, the latent heat of phase transition continues to evolve. If the condenser is sufficiently long, one more nucleation zone may appear. As was noted above, the size of growing particles depends on the place of their origination: droplets may grow to different sizes even if they travel the same distance along the chamber axis. Figure 9 shows the clus-
ter growth when the chamber operates at a high nucleation rate (the total number of particles in a unit volume of the LDFC is 2 × 1010). At the center of the channel, where the vapor density is maximal, the supersaturation drops and the diffusion growth of droplets slows down substantially as a result of overheating. The convergence of curves 1 and 2 indicates that the droplets grow by the diffusion mechanism discovered in [10]. At high degrees of supersaturation, the effects of local structures in the supersaturation fields begin to play a decisive role. These effects were qualitatively estimated, which allowed us to explain a number of paradoxical experimental results obtained in [6]. Quantitative characterization of these effects will be the subject of subsequent research.
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ACKNOWLEDGMENTS We thank S.I. Shabunya for valuable consultations. REFERENCES 1. A. A. Brin and S. P. Fisenko, Zh. Tekh. Fiz. 76 (4), 26 (2006) [Tech. Phys. 51 (4), 418 (2006)]. 2. B. S. Petukhov, Heat Exchange and Resistance in Laminar Flows through Tubes (Énergiya, Moscow, 1967) [in Russian]. 3. L. D. Landau and E. M. Lifshitz, Fluid Mechanics, 2nd ed. (Nauka, Moscow, 1986; Pergamon, Oxford, 1987). 4. V. M. Verzhbitskiœ, Fundamentals of Numerical Methods (Vysshaya Shkola, Moscow, 2002) [in Russian]. 5. J. H. Ferziger and H. G. Kaper, Mathematical Theory of Transport Processes in Gases (North Holland, Amsterdam, 1972). 6. V. Vohra and R. Heist, J. Chem. Phys. 104, 382 (1996). 7. N. B. Vargaftik, Tables of the Thermophysical Properties of Liquids and Gases (Nauka, Moscow, 1972; Halsted, New York, 1975).
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8. R. C. Reid, J. M. Prausnitz, and B. E. Poling, The Properties of Gases and Liquids (McGraw-Hill, New York, 1987). 9. Ya. I. Frenkel, Kinetic Theory of Liquids (Nauka, Leningrad, 1975; Clarendon, Oxford, 1946). 10. Ya. Vitovets, N. V. Pavlyukevich, S. P. Fisenko, et al., Inzh.-Fiz. Zh. 56, 936 (1989). 11. S. P. Fisenko and R. Heist, in Proceedings of the 6th Research Workshop on Nucleation Theory and Applications, Dubna, 2002, pp. 146–164.
12. S. P. Fisenko and G. Wilemski, Phys. Rev. E 70, 056119 (2004). 13. E. M. Lifshitz and L. P. Pitaevskii, Physical Kinetics (Nauka, Moscow, 1979; Pergamon, Oxford, 1981). 14. G. Madras and B. J. McCoy, J. Chem. Phys. 115, 6699 (2001). 15. J. Wedekind, K. Iland, P. E. Wagner, et al., in Proceedings of the 16th International Conference on Nucleation and Atmospheric Aerosols, Kyoto, 2004, pp. 49–52.
Translated by K. Shakhlevich
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