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Homogeneous nucleation rate measurements in 1-pentanol vapor with helium as a buffer gas
Atmospheric Research 46 Ž1998. 391–400
Homogeneous nucleation rate measurements in 1-pentanol vapor with helium as a buffer gas ) ˇ ´ Vladimır , Jirı ´ Zdımal ˇ´ Smolık ´ Institute of Chemical Process Fundamentals, Academy of Science of the Czech Republic, RozÕojoÕa´ 135, 16502 Prague, Czech Republic
Abstract The rate of homogeneous nucleation in supersaturated vapors of n-pentanol was studied experimentally using an upward static diffusion chamber. Helium was used as a buffer gas, holding the total pressure in the chamber at Pt s 25 kPa. A recently improved photographic technique was used to determine the nucleation rate as a function of supersaturation at temperature T s 260 K. This dependence was compared with predictions made by the classical theory of homogeneous nucleation. Furthermore, the influence of gaseous ions on nucleation rate was studied, and a minimum voltage across the chamber, necessary to avoid nucleation on ions, was determined. The effect of the wall heating power on nucleation was found to be negligible in the range studied. q 1998 Elsevier Science B.V. Keywords: Homogeneous nucleation; n-Pentanol; Helium
1. Introduction Nucleation is an integral part of vapor to condensed phase transitions. If nucleating sites are not present, condensation takes place by vapor deposition on its own embryos. This process, termed homogeneous nucleation, represents the simplest system for both experimental and theoretical investigations of nucleation in vapor. The experimental techniques used for this purpose are reviewed by Heist and He Ž1994.. Among them, static diffusion chambers are widely used, both for determination of critical supersaturation and for nucleation rate measurements. In these measurements, the nucleation rate is usually derived from the integral flux of droplets through an )
Corresponding author.
0169-8095r98r$19.00 q 1998 Elsevier Science B.V. All rights reserved. PII S 0 1 6 9 - 8 0 9 5 Ž 9 7 . 0 0 0 7 7 - X
ˇ´ V. Zdımal, J. Smolıkr ´ Atmospheric Research 46 (1998) 391–400
392
optical counter. The variation of supersaturation and temperature between the lower and upper chamber plates and the method of detection create difficulties in the comparison of results with results obtained from other techniques. Recently, the method of detection of nucleation in the chamber was modified by ˇ ´ Ž1994.. This new method allows the determination of nucleation rate Smolık ´ and Zdımal as a function of temperature and supersaturation while being independent of any nucleation theory. Here, this technique is used to measure the homogeneous nucleation rate in supersaturated vapors of n-pentanol in helium at temperature 260 K. It enables us to qualitatively compare our data with other pentanol data obtained by Strey et al. Ž1986. in a two piston expansion chamber. Unfortunately, another set of data obtained in a high pressure static diffusion chamber by Chukanov and Korobitsyn Ž1989., starts at conditions where our chamber cannot be operated.
2. Experimental The static diffusion chamber used in this research consists of two copper plates separated by a 162 mm i.d. optical glass ring, 25 mm in height, sealed in plates by Viton sealing. The inner surface of both plates is covered by a 10 m m thick golden sheet. Calibrated thermocouples are used to measure the temperature of both plates and a pressure transducer measures the total pressure inside the chamber. The bottom plate carrying a shallow pool Žapproximately 1 mm deep. of the investigated liquid is heated, vapor diffuses through a stagnant buffer gas and condenses on the cooler top plate. The top plate is slightly conical, so that the condensate flows to its edge and along the glass wall back to the pool. If the amount of buffer gas is properly selected, so that convection is avoided and the chamber operates at the steady state, temperature T and partial vapor pressure Pv decrease almost linearly from the bottom to the top. Since, the equilibrium vapor pressure Peq is approximately an exponential function of temperature, it decreases with the height of the chamber more rapidly than the actual vapor pressure. The vapor in the chamber therefore becomes supersaturated. Supersaturation, S Ždefined as the ratio of partial to equilibrium vapor pressure, S s PvrPeq ., has its maximum close to the top plate of the chamber. The profiles of temperature and partial vapor pressure Žand hence supersaturation. are found through the solution of heat and mass transfer equations ŽKatz, 1970. d Pv dz dT dz
s
Ž Pv y Pt . Nv T
½
s
0 Dvg
s ly1 vg yQ q Nv
q
a vg Pv Ž Pv y Pt . dT TPt T
HT C 2
P ,v dT q D H v
Ž 1.
dz
Ž T2 . q a vg RT
Ž Pv y Pt . Pt
5
Ž 2.
where Pv and Pt are the partial vapor pressure and total pressure, respectively, s and 0 Dvg are parameters of the temperature dependence of the binary diffusion coefficient, Nv is the molar flux density of the vapor, a vg is the thermal diffusion factor, l vg is the thermal conductivity of the vapor–gas mixture, Q is the heat flux density, CP ,v is the
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393
molar heat capacity of the vapor, D Hv is the enthalpy of vaporisation, R is the universal gas constant, and z is the length coordinate. These equations are solved numerically assuming the following boundary conditions: z s 0 T s T1 Pv s Pv ,eq Ž T1 . z s h T s T2 Pv s Pv ,eq Ž T2 .
Ž 3.
where h is the height of the chamber, T1 is the temperature of the liquid pool surface, T2 is the temperature of the top plate corrected for the liquid film thickness ŽKatz, 1970. and Pv,eq is the equilibrium vapor pressure at the temperature of the surface, corrected for mass flux ŽBecker, 1980.. By increasing the temperature difference between the plates one can increase supersaturation gradually and arrange the state in which supersaturation is sufficient for homogeneous nucleation to begin. The droplets, once formed by nucleation, grow rapidly to the visible size and fall down to the liquid pool. To observe nucleation in the chamber, the interior of the chamber is illuminated by a flat vertical laser beam passing through its centre. Trajectories of the droplets, formed inside this beam, are visible and can be photographed using a camera positioned exactly perpendicular to the beam. Each droplet photographed is characterised by its starting point Žpoint of its origin. and its trajectory ending in the liquid pool at the bottom plate. After evaluating enough particles Žstarting points. in one experiment, one gets the distribution of homogeneous nucleation rates as a function of height in the chamber. The measured distribution is then related to the corresponding values of temperature and supersaturation. This procedure can be seen in Fig. 1. On the left side of this figure, droplet trajectories are shown as they appear on photographs. In the three windows on the right there are local values of nucleation rate J as determined from photographs, and calculated profiles of temperature T and supersaturation S. This photographic method of evaluation was described in detail by Smolık ´ and ˇ ´ Ž1994.. It has three major advantages: Ža. it allows the determination of Zdımal nucleation rate as a function of temperature and supersaturation while being independent of any nucleation theory; Žb. it makes possible to exclude those experiments influenced by convection; Žc. it enables us to judge the influence of thermo- and diffusiophoresis.
Fig. 1. Nucleation rate as a function of temperature and supersaturation—method of evaluation.
394
ˇ´ V. Zdımal, J. Smolıkr ´ Atmospheric Research 46 (1998) 391–400
3. Substances and properties Purity of substances under investigation is as follows: n-pentanol p.a. ŽMerck Germany, catalogue No. 100975. was used without further purification; helium ŽLinde Technoplyn, Czech Republic. was in purity 99.999%. The physical parameters needed for calculation of profiles of temperature and supersaturation in the chamber, using Eqs. Ž1. – Ž3. include the thermal conductivity, the thermal diffusion factor, and the binary diffusion coefficient for the vapor–gas mixture, the molar heat capacity and the equilibrium vapor pressure of the 1-pentanol vapor. In order to calculate the dependence of vapor–gas mixture thermal conductivity on composition and temperature, the thermal conductivity and viscosity of both components are needed. The thermal conductivity and viscosity of helium were taken from the paper by Hung et al. Ž1989. and the thermal conductivity of 1-pentanol vapor from the paper by Shushpanov Ž1939. and a book by Vargaftik et al. Ž1978.. Further, there is some data on binary diffusion coefficient of 1-pentanol vapor in air ŽWinkelmann, 1885; Lugg, 1968., helium ŽKaraiskakis and Katsanos, 1984; Seager et al., 1963., hydrogen ŽWinkelmann, 1885., carbon dioxide ŽWinkelmann, 1885. and nitrogen ŽStepanov, 1981.. The Chapman–Enskog theory ŽHirschfelder et al., 1954. has been used to estimate the viscosity and thermal diffusion factor of 1-pentanol vapor. These estimations require the Lennard–Jones parameters s and ´rk, that are usually found from viscosity. Since, the experimental data on 1-pentanol vapor viscosity is not available, data on binary diffusion coefficients has been used in its place. Using the downhill simplex method as described by Press et al. Ž1992., we have found s s 6.677e y 10 m and ´rk s 304.1 K. The comparison of experimental binary diffusion coefficients with predictions made by the Chapman–Enskog theory Žusing estimated Lennard–Jones parameters. is shown in
Fig. 2. The comparison of experimental binary diffusion coefficients with predictions made by the Chapman– Enskog theory Žusing estimated Lennard–Jones parameters..
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Fig. 2. As can be seen, such predictions are very close to the available experimental data. Further, we have compared the experimental thermal conductivity of 1-pentanol vapor with predictions made by the modified Eucken correlation for polyatomic gases ŽReid et al., 1977., again using Lennard–Jones parameters calculated from diffusivities. The deviations of the predicted values from the experimental data did not exceed 5%. This leads us to the conclusion that the Chapman–Enskog theory, with Lennard–Jones parameters derived from diffusivities, gives satisfactory predictions of necessary transport coefficients. The final relationships used in calculations of the transport properties, the molar heat capacity ŽReid et al., 1977. and vapor pressure of 1-pentanol ŽSchmelling and Strey, 1983. are given in Table 1.
Table 1 Physical parameters needed for calculation of temperature and supersaturation profiles in the chamber n-Pentanol M s88.150 s s6.677=10y1 0 ´ r k s 304.1 l s1.86=10y2 y9.068=10y5 T q2.456=10y7 T 2 h s 2.6693=10y2 6 Ž MT . 0.5 r s 2V Ž2.2. Ž kT r ´ . CP ,v s 3.8686q5.0451=10y1 T y2.6394=10y4 T 2 q5.120=10y8 T 3 Peq s133.322expŽ90.079043y9788.384r T y9.9logŽT .. Helium M s 4.0026 s s 2.551=10y1 0 ´ r k s10.22 l sy2.45108=10y2 q1.12460=10y3 T y2.93123=10y6 T 2 q4.49646=10y9 T 3 y2.51948=10y1 2 T 4 h s1.4083.10y6 T 1.5 rŽT q70.22. n-Pentanol (Õ) – Helium (g) svg s 4.614=10y1 0 ´ vg r k s 55.75 l vg s x v l v rŽ x v q A vg x g .q x g lg rŽ x g q A gv x v . Ž1.1. Ž Dvg s 2.663=10y22 T 1.5 ŽŽ M v q Mg .rŽ2 M v Mg .. 0.5 rŽ Pt svg2 V vg kT r ´ vg .. 1r a vg s Žy0.7272yT rŽ16.36y0.2882T ..Ž x v q0.12281.q0.089303 M: Molar mass in g moley1 . s and ´ r k: Force constants for the Lennard–Jones Ž6–12. potential in m and K, respectively. k: Boltzmann constant in J Ky1 . l: Thermal conductivity of the gas in W my1 Ky1 . h : Viscosity of the gas in Pa s. V Ž i, j. : Collision integral of the type Ž i, j . as defined in Hirschfelder et al. Ž1954.. CP ,v : Constant pressure ideal-gas heat capacity of the vapor in J moly1 Ky1 . Peq : Equilibrium vapor pressure in Pa. x: Mole fraction. A vg and A gv : Factors in the Wassiljewa equation estimated by Mason–Saxena method ŽReid et al., 1977.. Dvg : Binary diffusion coefficient of n-pentanol in helium in m2 sy1 . a vg : Thermal diffusion factor calculated through Chapman–Enskog theory and correlated by the equation suggested by Kokugan and Shimizu Ž1981..
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4. Results and discussion At low rates of nucleation the homogeneous nucleation mechanism can compete with nucleation on ions resulting from natural radioactive sources or cosmic rays. For this reason an adjustable dc voltage supply was used to apply an electric field across the chamber. To determine the effect of electric field we measured the integral nucleation rate as a function of the potential applied. The results for two such experiments are shown in Fig. 3. It can be seen, that after a sharp decrease in observed nucleation rate a plateau is reached near a potential, E s 20 Vdcrcm. All of the following experiments were carried out at E s 25 Vdcrcm. There is another experimental difficulty related to the presence of chamber walls. These walls are slightly heated Žto avoid condensation on them. by resistance wires. The applied approach is based on the assumption that mass and heat transport Žin the central photographed part of the chamber. is described satisfactorily by the one-dimensional model. The absence of convection can be checked by examining droplet trajectories. However, it is necessary to determine a particular range of wall heating power, where the nucleation rate is independent on its change and the walls are free of condensate to allow photographing. The influence of the wall heating power on nucleation rate was studied in the whole range available. This range started at the point where last droplets of condensate just disappeared Ž0.17 Wrcm2 . and ended when the cryostat was unable to absorb any more heat, resulting in the collapse of its temperature control Ž0.52 Wrcm2 .. Both temperatures and the total pressure were kept constant during this experiment. In the whole range mentioned the nucleation rate varied within 16%, which is well inside the usual experimental precision. The photographic technique described previously was then used to determine the dependence of homogeneous nucleation rate on supersaturation at T s 260 K. It means,
Fig. 3. Dependence of the measured nucleation rate on the applied electric field.
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397
Fig. 4. Variation of nucleation rate with chamber height, fitted by Gaussian distribution.
that in all experiments, temperatures of the plates were chosen so that the isotherm 260 K lay near the centre of the nucleation zone. As was explained above, the solution of transfer equations gives the temperature and supersaturation at any height of the chamber but the experiments yield local values of the nucleation rate. Therefore, to obtain the variation of the rate of nucleation with the height of the chamber the local
Table 2 Raw data and results of the nucleation experiments Tb
Tt
Pt
S
Jexp
299.11 299.37 298.65 299.50 299.59 299.78 299.86 299.75 299.83 298.89 298.34 297.52 297.63
250.01 249.95 249.88 249.57 250.04 250.04 250.75 251.07 250.58 249.97 249.33 248.47 250.01
25.34 24.90 24.89 25.09 24.93 24.95 24.74 24.76 24.73 24.66 24.92 25.03 25.26
7.79 7.95 7.66 8.21 7.99 8.07 7.71 7.47 7.80 7.73 7.82 7.86 7.16
4.1 11.1 1.7 23.1 15.7 20.9 2.2 0.84 2.9 1.8 3.0 3.2 0.32
T b : Bottom plate temperature in K. Tt : Top plate temperature in K. Pt : Total pressure in kPa. S: Supersaturation at that height in the chamber, where T s 260 K Žcalculated value.. Jexp : Homogeneous nucleation rate corresponding to that height in the chamber, where T s 260 K Žfound from Gaussian fit of the experimentally determined nucleation rate distribution..
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398
Fig. 5. Experimental nucleation rates in comparison with theoretical prediction.
nucleation rates, determined experimentally, were fitted by Gaussian distribution function. The typical fit is shown in Fig. 4, where the upper and lower dotted curves correspond to a 95% confidence level. The rate of nucleation at T s 260 K was then obtained using this fit. The raw data and results of the experiments are given in Table 2. The resulting dependence of homogeneous nucleation rate on supersaturation at constant nucleation temperature T s 260 K is shown in Fig. 5. Here, the vertical error bars denote the uncertainty of the Gaussian fit. Horizontal error bars show the uncertainty in supersaturation caused by slight variation in temperatures of the chamber plates during the experiment Žalways less than "0.1 K during 6 h.. This relatively large uncertainty is calculated for the less favourite combination of errors in determination of temperatures. It can be seen, that the experimental slope is in close agreement with that of classical theory of homogeneous nucleation. However, the experimental points are more than two orders of magnitude higher than the theoretical curve. Such a result from nucleation experiments is not unusual and is in qualitative agreement with earlier expansion chamber results ŽStrey et al., 1986..
5. Conclusions The rate of homogeneous nucleation in supersaturated n-pentanol vapor was studied experimentally using a static diffusion chamber. The dependence of nucleation rate on supersaturation was determined at a temperature of 260 K. It was found, that the experimental slope is in qualitative agreement with that predicted by the classical theory of homogeneous nucleation and with previous results obtained by Strey et al. Ž1986.. The experimental points are more than two orders of magnitude higher than the theoretical curve. The influence of gaseous ions on nucleation rate was also studied and
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a minimum voltage across the chamber, necessary to avoid nucleation on ions, was determined. Further, a study concerning optimum wall heating power was carried out. It was found, that the nucleation rate does not depend on the wall heating power in the studied range.
6. Correction An error in the extended abstract submitted to the European Aerosol Conference 1996 in Helsinki was found. In the figure, showing the variation of the measured nucleation rate with supersaturation, all of our experimental data points are to be increased by a factor of 15. This error arose during the numerical evaluation of the experiments.
Acknowledgements Support of this work by the grant No. A4072504 of the Grant agency AS CR is gratefully acknowledged. The authors wish to thank Mr. Shane P. Morrin for his help in performing the experimental work.
References Becker, C., 1980. Homogeneous vapor phase nucleation of a large fatty acid: stearic acid. J. Chem. Phys. 72, 4579–4587. Chukanov, V.N., Korobitsyn, B.A., 1989. Kinetika statsionarnoi gomogennoi nukleatsii v peresyshchennom pare Žin Russian.. Dokl. Akad. Nauk USSR 307, 153–156. Heist, R.H., He, H., 1994. Review of vapor to liquid homogeneous nucleation experiments from 1968 to 1992. J. Phys. Chem. Ref. Data 23, 781–805. Hirschfelder, J.O., Curtiss, C.F., Bird R.B., 1954. Molecular Theory of Gases and Liquids. Wiley, New York, 1219 pp. Hung, C.-H., Krasnopoler, M.J., Katz, J.L., 1989. Condensation of a supersaturated vapor: VIII. The homogeneous nucleation of n-nonane. J. Chem. Phys. 90, 1856–1865. Karaiskakis, G., Katsanos, N.A., 1984. Rate coefficients for evaporation of pure liquids and diffusion coefficients of vapors. J. Phys. Chem. 88, 3674–3678. Katz, J.L., 1970. Condensation of a supersaturated vapor: I. The homogeneous nucleation of the n-alkanes. J. Chem. Phys. 52, 4733–4748. Kokugan, T., Shimizu, M., 1981. Measurement of apparent thermal diffusion factor of three-component ŽHe–Ar–Kr. gas mixture. J. Chem. Eng. Jpn. 14, 7–12. Lugg, G.A., 1968. Diffusion coefficients of some organic and other vapors in air. Anal. Chem. 40, 1072–1077. Press, W.H., Teukolsky, S.A., Vetterling, W.T., Flannery, B.P., 1992. Numerical Recipes in C. The Art of Scientific Computing. Cambridge Univ. Press, New York, 994 pp. Reid, R.Q., Prausnitz, J.M., Sherwood, T.K., 1977. The Properties of Gases and Liquids, 3rd edn. McGraw-Hill, New York, 688 pp. Schmelling, T., Strey, R., 1983. Equilibrium vapor pressure measurements for the n-alcohols in the temperature range from y308C to 308C. Ber. Bunsenges. Phys. Chem. 87, 871–874. Seager, S.L., Geertson, L.R., Giddings, J.C., 1963. Temperature dependence of gas and vapor diffusion coefficients. J. Chem. Eng. Data 8, 168–169.
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Shushpanov, P.I., 1939. K voprosu o teploprovodnosti organitsheskikh soedinenii Žin Russian.. J. Exp. Theor. Phys. USSR 9, 875–883. ˇ ´ Smolık, V., 1994. Condensation of supersaturated vapors of dioctylphthalate. Homogeneous ´ J., Zdımal, nucleation rate measurements. Aerosol Sci. Technol. 20, 127–134. Stepanov, J.A., 1981. Koefficienty diffuzii parov riada odnoatomnykh spirtov v zzhatye gazy Žin Russian.. Teplo-Masoobmen Khim. Tekhnol., pp. 49–51. Strey, R., Wagner, P.E., Schmelling, T., 1986. Homogeneous nucleation rates for n-alcohol vapors measured in a two piston expansion chamber. J. Chem. Phys. 84, 2325–2335. Vargaftik, N.B., Filippov, L.P., Tarzimanov, A.A., Tockij, E.E., 1978. Teploprovodnost’ zhidkostei i gazov Žin Russian.. Moskva. Winkelmann, A., 1885. Ueber die Diffusion der Fettsauren und Fettalkohole in Luft, Wasserstoff und Kohlensaure. Ann. Phys. 26, 105–134.
Homogeneous nucleation rate measurements in 1-pentanol vapor with helium as a buffer gas ) ˇ ´ Vladimır , Jirı ´ Zdımal ˇ´ Smolık ´ Institute of Chemical Process Fundamentals, Academy of Science of the Czech Republic, RozÕojoÕa´ 135, 16502 Prague, Czech Republic
Abstract The rate of homogeneous nucleation in supersaturated vapors of n-pentanol was studied experimentally using an upward static diffusion chamber. Helium was used as a buffer gas, holding the total pressure in the chamber at Pt s 25 kPa. A recently improved photographic technique was used to determine the nucleation rate as a function of supersaturation at temperature T s 260 K. This dependence was compared with predictions made by the classical theory of homogeneous nucleation. Furthermore, the influence of gaseous ions on nucleation rate was studied, and a minimum voltage across the chamber, necessary to avoid nucleation on ions, was determined. The effect of the wall heating power on nucleation was found to be negligible in the range studied. q 1998 Elsevier Science B.V. Keywords: Homogeneous nucleation; n-Pentanol; Helium
1. Introduction Nucleation is an integral part of vapor to condensed phase transitions. If nucleating sites are not present, condensation takes place by vapor deposition on its own embryos. This process, termed homogeneous nucleation, represents the simplest system for both experimental and theoretical investigations of nucleation in vapor. The experimental techniques used for this purpose are reviewed by Heist and He Ž1994.. Among them, static diffusion chambers are widely used, both for determination of critical supersaturation and for nucleation rate measurements. In these measurements, the nucleation rate is usually derived from the integral flux of droplets through an )
Corresponding author.
0169-8095r98r$19.00 q 1998 Elsevier Science B.V. All rights reserved. PII S 0 1 6 9 - 8 0 9 5 Ž 9 7 . 0 0 0 7 7 - X
ˇ´ V. Zdımal, J. Smolıkr ´ Atmospheric Research 46 (1998) 391–400
392
optical counter. The variation of supersaturation and temperature between the lower and upper chamber plates and the method of detection create difficulties in the comparison of results with results obtained from other techniques. Recently, the method of detection of nucleation in the chamber was modified by ˇ ´ Ž1994.. This new method allows the determination of nucleation rate Smolık ´ and Zdımal as a function of temperature and supersaturation while being independent of any nucleation theory. Here, this technique is used to measure the homogeneous nucleation rate in supersaturated vapors of n-pentanol in helium at temperature 260 K. It enables us to qualitatively compare our data with other pentanol data obtained by Strey et al. Ž1986. in a two piston expansion chamber. Unfortunately, another set of data obtained in a high pressure static diffusion chamber by Chukanov and Korobitsyn Ž1989., starts at conditions where our chamber cannot be operated.
2. Experimental The static diffusion chamber used in this research consists of two copper plates separated by a 162 mm i.d. optical glass ring, 25 mm in height, sealed in plates by Viton sealing. The inner surface of both plates is covered by a 10 m m thick golden sheet. Calibrated thermocouples are used to measure the temperature of both plates and a pressure transducer measures the total pressure inside the chamber. The bottom plate carrying a shallow pool Žapproximately 1 mm deep. of the investigated liquid is heated, vapor diffuses through a stagnant buffer gas and condenses on the cooler top plate. The top plate is slightly conical, so that the condensate flows to its edge and along the glass wall back to the pool. If the amount of buffer gas is properly selected, so that convection is avoided and the chamber operates at the steady state, temperature T and partial vapor pressure Pv decrease almost linearly from the bottom to the top. Since, the equilibrium vapor pressure Peq is approximately an exponential function of temperature, it decreases with the height of the chamber more rapidly than the actual vapor pressure. The vapor in the chamber therefore becomes supersaturated. Supersaturation, S Ždefined as the ratio of partial to equilibrium vapor pressure, S s PvrPeq ., has its maximum close to the top plate of the chamber. The profiles of temperature and partial vapor pressure Žand hence supersaturation. are found through the solution of heat and mass transfer equations ŽKatz, 1970. d Pv dz dT dz
s
Ž Pv y Pt . Nv T
½
s
0 Dvg
s ly1 vg yQ q Nv
q
a vg Pv Ž Pv y Pt . dT TPt T
HT C 2
P ,v dT q D H v
Ž 1.
dz
Ž T2 . q a vg RT
Ž Pv y Pt . Pt
5
Ž 2.
where Pv and Pt are the partial vapor pressure and total pressure, respectively, s and 0 Dvg are parameters of the temperature dependence of the binary diffusion coefficient, Nv is the molar flux density of the vapor, a vg is the thermal diffusion factor, l vg is the thermal conductivity of the vapor–gas mixture, Q is the heat flux density, CP ,v is the
ˇ´ V. Zdımal, J. Smolıkr ´ Atmospheric Research 46 (1998) 391–400
393
molar heat capacity of the vapor, D Hv is the enthalpy of vaporisation, R is the universal gas constant, and z is the length coordinate. These equations are solved numerically assuming the following boundary conditions: z s 0 T s T1 Pv s Pv ,eq Ž T1 . z s h T s T2 Pv s Pv ,eq Ž T2 .
Ž 3.
where h is the height of the chamber, T1 is the temperature of the liquid pool surface, T2 is the temperature of the top plate corrected for the liquid film thickness ŽKatz, 1970. and Pv,eq is the equilibrium vapor pressure at the temperature of the surface, corrected for mass flux ŽBecker, 1980.. By increasing the temperature difference between the plates one can increase supersaturation gradually and arrange the state in which supersaturation is sufficient for homogeneous nucleation to begin. The droplets, once formed by nucleation, grow rapidly to the visible size and fall down to the liquid pool. To observe nucleation in the chamber, the interior of the chamber is illuminated by a flat vertical laser beam passing through its centre. Trajectories of the droplets, formed inside this beam, are visible and can be photographed using a camera positioned exactly perpendicular to the beam. Each droplet photographed is characterised by its starting point Žpoint of its origin. and its trajectory ending in the liquid pool at the bottom plate. After evaluating enough particles Žstarting points. in one experiment, one gets the distribution of homogeneous nucleation rates as a function of height in the chamber. The measured distribution is then related to the corresponding values of temperature and supersaturation. This procedure can be seen in Fig. 1. On the left side of this figure, droplet trajectories are shown as they appear on photographs. In the three windows on the right there are local values of nucleation rate J as determined from photographs, and calculated profiles of temperature T and supersaturation S. This photographic method of evaluation was described in detail by Smolık ´ and ˇ ´ Ž1994.. It has three major advantages: Ža. it allows the determination of Zdımal nucleation rate as a function of temperature and supersaturation while being independent of any nucleation theory; Žb. it makes possible to exclude those experiments influenced by convection; Žc. it enables us to judge the influence of thermo- and diffusiophoresis.
Fig. 1. Nucleation rate as a function of temperature and supersaturation—method of evaluation.
394
ˇ´ V. Zdımal, J. Smolıkr ´ Atmospheric Research 46 (1998) 391–400
3. Substances and properties Purity of substances under investigation is as follows: n-pentanol p.a. ŽMerck Germany, catalogue No. 100975. was used without further purification; helium ŽLinde Technoplyn, Czech Republic. was in purity 99.999%. The physical parameters needed for calculation of profiles of temperature and supersaturation in the chamber, using Eqs. Ž1. – Ž3. include the thermal conductivity, the thermal diffusion factor, and the binary diffusion coefficient for the vapor–gas mixture, the molar heat capacity and the equilibrium vapor pressure of the 1-pentanol vapor. In order to calculate the dependence of vapor–gas mixture thermal conductivity on composition and temperature, the thermal conductivity and viscosity of both components are needed. The thermal conductivity and viscosity of helium were taken from the paper by Hung et al. Ž1989. and the thermal conductivity of 1-pentanol vapor from the paper by Shushpanov Ž1939. and a book by Vargaftik et al. Ž1978.. Further, there is some data on binary diffusion coefficient of 1-pentanol vapor in air ŽWinkelmann, 1885; Lugg, 1968., helium ŽKaraiskakis and Katsanos, 1984; Seager et al., 1963., hydrogen ŽWinkelmann, 1885., carbon dioxide ŽWinkelmann, 1885. and nitrogen ŽStepanov, 1981.. The Chapman–Enskog theory ŽHirschfelder et al., 1954. has been used to estimate the viscosity and thermal diffusion factor of 1-pentanol vapor. These estimations require the Lennard–Jones parameters s and ´rk, that are usually found from viscosity. Since, the experimental data on 1-pentanol vapor viscosity is not available, data on binary diffusion coefficients has been used in its place. Using the downhill simplex method as described by Press et al. Ž1992., we have found s s 6.677e y 10 m and ´rk s 304.1 K. The comparison of experimental binary diffusion coefficients with predictions made by the Chapman–Enskog theory Žusing estimated Lennard–Jones parameters. is shown in
Fig. 2. The comparison of experimental binary diffusion coefficients with predictions made by the Chapman– Enskog theory Žusing estimated Lennard–Jones parameters..
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Fig. 2. As can be seen, such predictions are very close to the available experimental data. Further, we have compared the experimental thermal conductivity of 1-pentanol vapor with predictions made by the modified Eucken correlation for polyatomic gases ŽReid et al., 1977., again using Lennard–Jones parameters calculated from diffusivities. The deviations of the predicted values from the experimental data did not exceed 5%. This leads us to the conclusion that the Chapman–Enskog theory, with Lennard–Jones parameters derived from diffusivities, gives satisfactory predictions of necessary transport coefficients. The final relationships used in calculations of the transport properties, the molar heat capacity ŽReid et al., 1977. and vapor pressure of 1-pentanol ŽSchmelling and Strey, 1983. are given in Table 1.
Table 1 Physical parameters needed for calculation of temperature and supersaturation profiles in the chamber n-Pentanol M s88.150 s s6.677=10y1 0 ´ r k s 304.1 l s1.86=10y2 y9.068=10y5 T q2.456=10y7 T 2 h s 2.6693=10y2 6 Ž MT . 0.5 r s 2V Ž2.2. Ž kT r ´ . CP ,v s 3.8686q5.0451=10y1 T y2.6394=10y4 T 2 q5.120=10y8 T 3 Peq s133.322expŽ90.079043y9788.384r T y9.9logŽT .. Helium M s 4.0026 s s 2.551=10y1 0 ´ r k s10.22 l sy2.45108=10y2 q1.12460=10y3 T y2.93123=10y6 T 2 q4.49646=10y9 T 3 y2.51948=10y1 2 T 4 h s1.4083.10y6 T 1.5 rŽT q70.22. n-Pentanol (Õ) – Helium (g) svg s 4.614=10y1 0 ´ vg r k s 55.75 l vg s x v l v rŽ x v q A vg x g .q x g lg rŽ x g q A gv x v . Ž1.1. Ž Dvg s 2.663=10y22 T 1.5 ŽŽ M v q Mg .rŽ2 M v Mg .. 0.5 rŽ Pt svg2 V vg kT r ´ vg .. 1r a vg s Žy0.7272yT rŽ16.36y0.2882T ..Ž x v q0.12281.q0.089303 M: Molar mass in g moley1 . s and ´ r k: Force constants for the Lennard–Jones Ž6–12. potential in m and K, respectively. k: Boltzmann constant in J Ky1 . l: Thermal conductivity of the gas in W my1 Ky1 . h : Viscosity of the gas in Pa s. V Ž i, j. : Collision integral of the type Ž i, j . as defined in Hirschfelder et al. Ž1954.. CP ,v : Constant pressure ideal-gas heat capacity of the vapor in J moly1 Ky1 . Peq : Equilibrium vapor pressure in Pa. x: Mole fraction. A vg and A gv : Factors in the Wassiljewa equation estimated by Mason–Saxena method ŽReid et al., 1977.. Dvg : Binary diffusion coefficient of n-pentanol in helium in m2 sy1 . a vg : Thermal diffusion factor calculated through Chapman–Enskog theory and correlated by the equation suggested by Kokugan and Shimizu Ž1981..
396
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4. Results and discussion At low rates of nucleation the homogeneous nucleation mechanism can compete with nucleation on ions resulting from natural radioactive sources or cosmic rays. For this reason an adjustable dc voltage supply was used to apply an electric field across the chamber. To determine the effect of electric field we measured the integral nucleation rate as a function of the potential applied. The results for two such experiments are shown in Fig. 3. It can be seen, that after a sharp decrease in observed nucleation rate a plateau is reached near a potential, E s 20 Vdcrcm. All of the following experiments were carried out at E s 25 Vdcrcm. There is another experimental difficulty related to the presence of chamber walls. These walls are slightly heated Žto avoid condensation on them. by resistance wires. The applied approach is based on the assumption that mass and heat transport Žin the central photographed part of the chamber. is described satisfactorily by the one-dimensional model. The absence of convection can be checked by examining droplet trajectories. However, it is necessary to determine a particular range of wall heating power, where the nucleation rate is independent on its change and the walls are free of condensate to allow photographing. The influence of the wall heating power on nucleation rate was studied in the whole range available. This range started at the point where last droplets of condensate just disappeared Ž0.17 Wrcm2 . and ended when the cryostat was unable to absorb any more heat, resulting in the collapse of its temperature control Ž0.52 Wrcm2 .. Both temperatures and the total pressure were kept constant during this experiment. In the whole range mentioned the nucleation rate varied within 16%, which is well inside the usual experimental precision. The photographic technique described previously was then used to determine the dependence of homogeneous nucleation rate on supersaturation at T s 260 K. It means,
Fig. 3. Dependence of the measured nucleation rate on the applied electric field.
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397
Fig. 4. Variation of nucleation rate with chamber height, fitted by Gaussian distribution.
that in all experiments, temperatures of the plates were chosen so that the isotherm 260 K lay near the centre of the nucleation zone. As was explained above, the solution of transfer equations gives the temperature and supersaturation at any height of the chamber but the experiments yield local values of the nucleation rate. Therefore, to obtain the variation of the rate of nucleation with the height of the chamber the local
Table 2 Raw data and results of the nucleation experiments Tb
Tt
Pt
S
Jexp
299.11 299.37 298.65 299.50 299.59 299.78 299.86 299.75 299.83 298.89 298.34 297.52 297.63
250.01 249.95 249.88 249.57 250.04 250.04 250.75 251.07 250.58 249.97 249.33 248.47 250.01
25.34 24.90 24.89 25.09 24.93 24.95 24.74 24.76 24.73 24.66 24.92 25.03 25.26
7.79 7.95 7.66 8.21 7.99 8.07 7.71 7.47 7.80 7.73 7.82 7.86 7.16
4.1 11.1 1.7 23.1 15.7 20.9 2.2 0.84 2.9 1.8 3.0 3.2 0.32
T b : Bottom plate temperature in K. Tt : Top plate temperature in K. Pt : Total pressure in kPa. S: Supersaturation at that height in the chamber, where T s 260 K Žcalculated value.. Jexp : Homogeneous nucleation rate corresponding to that height in the chamber, where T s 260 K Žfound from Gaussian fit of the experimentally determined nucleation rate distribution..
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398
Fig. 5. Experimental nucleation rates in comparison with theoretical prediction.
nucleation rates, determined experimentally, were fitted by Gaussian distribution function. The typical fit is shown in Fig. 4, where the upper and lower dotted curves correspond to a 95% confidence level. The rate of nucleation at T s 260 K was then obtained using this fit. The raw data and results of the experiments are given in Table 2. The resulting dependence of homogeneous nucleation rate on supersaturation at constant nucleation temperature T s 260 K is shown in Fig. 5. Here, the vertical error bars denote the uncertainty of the Gaussian fit. Horizontal error bars show the uncertainty in supersaturation caused by slight variation in temperatures of the chamber plates during the experiment Žalways less than "0.1 K during 6 h.. This relatively large uncertainty is calculated for the less favourite combination of errors in determination of temperatures. It can be seen, that the experimental slope is in close agreement with that of classical theory of homogeneous nucleation. However, the experimental points are more than two orders of magnitude higher than the theoretical curve. Such a result from nucleation experiments is not unusual and is in qualitative agreement with earlier expansion chamber results ŽStrey et al., 1986..
5. Conclusions The rate of homogeneous nucleation in supersaturated n-pentanol vapor was studied experimentally using a static diffusion chamber. The dependence of nucleation rate on supersaturation was determined at a temperature of 260 K. It was found, that the experimental slope is in qualitative agreement with that predicted by the classical theory of homogeneous nucleation and with previous results obtained by Strey et al. Ž1986.. The experimental points are more than two orders of magnitude higher than the theoretical curve. The influence of gaseous ions on nucleation rate was also studied and
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a minimum voltage across the chamber, necessary to avoid nucleation on ions, was determined. Further, a study concerning optimum wall heating power was carried out. It was found, that the nucleation rate does not depend on the wall heating power in the studied range.
6. Correction An error in the extended abstract submitted to the European Aerosol Conference 1996 in Helsinki was found. In the figure, showing the variation of the measured nucleation rate with supersaturation, all of our experimental data points are to be increased by a factor of 15. This error arose during the numerical evaluation of the experiments.
Acknowledgements Support of this work by the grant No. A4072504 of the Grant agency AS CR is gratefully acknowledged. The authors wish to thank Mr. Shane P. Morrin for his help in performing the experimental work.
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