- Email: [email protected]
Numerical analysis of gas-phase diffusion resistance in a droplet train apparatus
13 August 2002
Chemical Physics Letters 362 (2002) 56–62 www.elsevier.com/locate/cplett
Numerical analysis of gas-phase diffusion resistance in a droplet train apparatus Masakazu Sugiyama
a,1
, Seiichiro Koda a, Akihiro Morita
b,*
a
b
Department of Chemical System Engineering, School of Engineering, The University of Tokyo, 7-3-1 Hongo, Bunkyo-ku, Tokyo 113-8656, Japan Department of Chemistry, Graduate School of Science, Kyoto University, Kitashirakawa, Sakyo-ku, Kyoto 606-8502, Japan Received 4 May 2002; in final form 6 June 2002
Abstract In this Letter, we theoretically analyze gaseous flow in a droplet train apparatus, which has been extensively utilized to study mass accommodation processes through gas/liquid interfaces. We numerically evaluate gas-phase resistance for uptake of trace gas into the droplets, via the coupled diffusion equation and fluid dynamics. In the results, gaseous resistance shows systematic deviation from values estimated from the ideal model with static spherical boundary conditions, indicating generation of additional resistance by interference among the train of droplets. Improved description of gas-phase resistance with optimized parameters in the flow tube is provided. Ó 2002 Elsevier Science B.V. All rights reserved.
1. Introduction Gas/liquid interfacial mass exchange is of critical significance in heterogeneous atmospheric chemistry and cloud microphysics [1–3]. The microscopic probability for interfacial mass transfer is characterized by the mass accommodation coefficient a, a ¼ðnumber of gas molecules absorbed into the liquidÞ =ðnumber of gas molecules impinging on the surfaceÞ;
ð1Þ *
Corresponding author. Fax: +81-75-753-4000. E-mail address: [email protected] (A. Morita). 1 Present address: Department of Electronics Engineering, School of Engineering, The University of Tokyo, 7-3-1 Hongo, Bunkyo-ku, Tokyo 113-8656, Japan.
which ranges from 0 to 1. However, experimentally determining the mass accommodation coefficient a from the phenomenological mass transfer rates [3,4] is not straightforward, since in most cases the interfacial mass exchange rates, characterized by uptake coefficients c, are influenced by numerous other factors, including gas-phase diffusion, solubility in the liquid phase, and reaction rate in/on the liquid. When chemical reactions in/ on the liquid are neglected, overall resistance of mass transfer 1=c is well described by the following decoupled form [4,5], 1 1 1 1 ¼ þ þ ; c Cg a Csol
ð2Þ
where the notations are as described in [4]; 1=Cg represents resistance of the gas-phase diffusion,
0009-2614/02/$ - see front matter Ó 2002 Elsevier Science B.V. All rights reserved. PII: S 0 0 0 9 - 2 6 1 4 ( 0 2 ) 0 1 0 1 9 - 9
M. Sugiyama et al. / Chemical Physics Letters 362 (2002) 56–62
and 1=Csol and represents the resistance of liquidphase solubility. Therefore, except for conditions where 1=Cg and 1=Csol are negligible, precise estimation of 1=Cg and 1=Csol is critical for experimental derivation of a. In the present Letter, we discuss accurate evaluation of the gas-phase resistance 1=Cg in a droplet train apparatus, since current methods of estimating 1=Cg might involve some uncertainty, as discussed below. The droplet train apparatus is one of the most widely employed experimental means for determining the mass accommodation coefficients [6– 10]. It utilizes a highly controlled train of droplets passing through a flow tube, and the uptake rate of the gaseous species into the droplets is determined from the reduction in the gaseous trace gas concentration after a certain gas–liquid interaction time. The droplets are generated by an orifice vibrating at a certain frequency and back pressure. The liquid surface is therefore continuously refreshed, and total surface area of the droplets is well controlled and varied by changing the orifice frequency. This apparatus utilizes relatively short and variable gas–liquid interaction time on the order of metersecond, thereby reducing the uncertainty in the solubility resistance 1=Csol . Droplet train experiments are usually conducted under low pressure conditions so as to minimize the gasphase resistance 1=Cg . These design features render this technique applicable within a wide range of the mass accommodation coefficients, including relatively large values up to unity. In the experimental analysis, the gas-phase resistance 1=Cg is usually evaluated via the Fuchs– Sutugin formula [11,12] or other alternative formulas [3]. The Fuchs–Sutugin formula describes the gas-phase resistance of uptake into a single spherical particle over a wide range of Knudsen number Kn, as follows 1 0:75 þ 0:283Kn ¼ Cg Knð1 þ KnÞ 0:75 0:467 ðfor low KnÞ; ð3Þ ! Kn where Kn ¼ 6Dg = cd; c is the average thermal vepffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi locity of the trace gas molecules ð¼ 8kB T =pM Þ; d the droplet diameter; and Dg the diffusion coefficient of the trace gas molecules. The limiting
57
formula of Eq. (3) in the continuum regime corresponds to the diffusive form of gas-phase resistance with the kinetic correction term 1/2 [4,13], 1 1 cd ¼ : Cg 8Dg 2
ð4Þ
The Fuchs–Sutugin formula Eqs. (3) or (4) involves an important assumption of the static spherical boundary condition around the droplet, whereas in an actual droplet train flow tube, a train of the droplets under flow should be considered [14]. Therefore, the purpose of this Letter is to examine the quantitative validity of Eq. (3) in droplet train flow tubes. We numerically solved the gaseous diffusion equation with fluid dynamics under flow conditions that correspond to the droplet train tube to thereby directly evaluate the gas-phase resistance. Accordingly, the present discussion focuses on the diffusive, continuum regime, which is often encountered under atmospheric conditions. The remainder of this Letter is constructed as follows. The next section explains the method and conditions employed for the calculations, and Section 3 provides results and discussion. Brief concluding remarks and future perspective follow in Section 4.
2. Calculation method and conditions Fig. 1 depicts the geometry of the boundary conditions. In a cylindrical flow tube having a diameter of 15 mm and a length of 20 mm, 50 spherical droplets are aligned on the axis at equal intervals. The droplet configuration is based on a typical set of experimental conditions; i.e. droplet velocity vd ¼ 27 m/s, orifice frequency fo ¼ 100 kHz, and orifice diameter do ¼ 60 lm [6]. Consequently, the center-of-mass distance between adjacent droplets is vd =fo ¼ 270 lm. The orifice diameter do ¼ 60 lm leads to the droplet diameter 1=3 d ¼ ð3do2 vd =2fo Þ 1:89do ¼ 113 lm via the liquid volume flow rate, F ¼
4p 3
3 2 d do fo ¼ p vd ; 2 2
ð5Þ
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Fig. 1. Cross-section of the cylindrical flow tube. Unit: mm. The wall and the gaseous flow move to the right in relation to the droplets.
and the Rayleigh natural breakup condition [6], fo ¼ vd =4:5do :
ð6Þ
The relative movement of the droplets to the wall was considered in actual calculations for fixed droplets. In Fig. 1, the wall of the flow tube is depicted as moving from left to right at a relative speed of vd with respect to the droplets. Calculations were performed under isothermal conditions within a temperature of )13–15 °C. Note that the droplets are supercooled but not frozen below 0 °C [6]. The gaseous volume flow rate is 200 cm3 =s, which means the gaseous average velocity is vg ¼ 200 cm3 =s=ðpð0:75 cmÞ2 Þ ¼ 1:13 m/ s. In order to incorporate this gaseous flow rate into the above boundary condition of fixed droplets and the moving tube wall, the average gaseous velocity at the inlet was set to vd vg for the same direction as the tube wall velocity. We assumed the sticking boundary condition at the surfaces of the wall and the droplets for the gaseous flow. The gas-phase species consists of a trace amount of methanol, saturated water vapor, and inert carrier gas of He or Ar. Initial methanol concentration is fixed as low as ng ¼ 1013 3 molecules=cm . For comparison, we also examined another concentration, 5 1013 molecules= cm3 , and confirmed the linearity between uptake rate and concentration. The saturated water vapor pressures are 1.7, 4.6, and 12.8 Torr at )13, 0, and
15 °C, respectively [4,15]. Total pressure ranges from 6 to 50 Torr at each temperature, with varying partial pressure of the carrier gas. The viscosity of the gas and the diffusion coefficients of the constituent molecules were evaluated by the Chapman–Enskog theory [16,17], where the collision integrals were calculated by the Stockmeyer model for polar molecules and by the Lennard– Jones model for nonpolar molecules. The ingredient parameters for the collision integrals were taken from literature [18]. This pressure range and the droplet diameter d assure the continuum description of the gas-phase diffusion, as the Knudsen number Kn K 0:2. In view of the low Reynolds numbers for the droplets, 0-5-10, we can safely assume laminar gaseous flow in the flow tube. Regarding the intrinsic deposition rate kint of the trace gas onto the droplet surface, we assumed two conditions-perfect uptake, and a finite rate kint ¼ c=2. The former condition of perfect uptake was actually implemented with a sufficiently large rate, kint ¼ 1035 cm=s 2, which effectively imposed vanishing concentration of the trace gas at the droplet surface. This is consistent with the usual Smoluchowski boundary condition [19]. Whereas the former condition completely neglects the intrinsic resistance of the mass accommodation, the 2
We compared another value kint ¼ 1035 cm=s to 1030 , and confirmed derivation of identical results, indicating that the present kint is sufficiently large.
M. Sugiyama et al. / Chemical Physics Letters 362 (2002) 56–62
latter condition effectively takes into account the finite collision rate [20]. The finite rate kint ¼ c=2 is twice the thermal collision rate c=4, corresponding to the kinetic correction term 1/2 in Eq. (4). Although the two assumptions yield little difference in the overall resistance in the continuum regime, we employed the latter assumption in order to discuss effective diameter and mass accommodation coefficient later. Numerical calculation of the coupled fluid and diffusion equations was performed by use of the F L U E N T package [21] in twodimensional cylindrical space. A total of 64,413 triangle mesh points were employed, including particularly fine radial mesh points near the gas/ droplet interface, which is the critical region for the uptake. To check the accuracy yielded by the mesh size, we used another finer mesh geometry, consisting of 110,626 quadrilateral grids, and confirmed that the calculated deposition rates for the two cell geometries agree within 0.5%.
59
3. Results and discussion Fig. 2 depicts steady-state gaseous velocity and trace gas concentration fields in the flow tube. The velocity field in Fig. 2a is nearly uniform except for the vicinity of the axis, where the strong perturbation is apparent, owing to the train of droplets. Relative gaseous velocity on the axis between the droplets is quite small, implying that the volume elements of the gas lying between the droplets pass along the axis at nearly the same speed as the droplets. This is consistent with experimental evidence that the train of the droplets suffers from little deceleration by frictional drag in the flow tube [22]. The concentration field of the trace gas in Fig. 2b shows substantial depletion in the vicinity of the droplet, train, and on the whole, cylindrical concentration distribution around the axis is developed rather than spherical distribution around each droplet. This feature of the emphasized radial
Fig. 2. Steady-state (a) gaseous velocity field and (b) concentration field of methanol in the flow tube at 0 °C and 25 Torr. Half cross sections of the cylinder tube are shown, where the top and bottom lines correspond to the wall and the axis, respectively. In panel (a), relative velocity of the cylinder wall with respect to the droplets is 27 m/s. In panel (b), contour spacing is ng =20. A magnified figure in the vicinity of the droplets is inserted in the upper right corner of panel (b).
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concentration gradient is attributed to the axial flow and to interference among the train of droplets. Since the axial concentration gradient becomes relatively unimportant with respect to the gaseous diffusion, the gas-phase diffusional transport approaches two-dimensional transport, implying deviation from Eqs. (3) or (4). The deposition rate of the trace gas methanol into the droplets per unit surface area, R, was calculated by the diffusion equation with the absorbing boundary at the droplet surface and compared with the ideal, spherical case. In the latter, ideal case, the steady-state Smoluchowski equation around a single spherical droplet of the diameter d yields a deposition rate Rideal ¼ 2Dg ng =d, which corresponds to the continuum regime of Eq. (4) without the kinetic correction term 1/2 and the normalization factor ng c=4; !, ! 1 d ng c cd ¼ ¼ : ð7Þ Rideal 2Dg ng 8Dg 4 Fig. 3 compares the calculated deposition rates R and the ideal deposition rates Rideal in the 1=Dg 1=R plot, where the latter should describe a linear
Fig. 3. Inverse deposition rate 1=R of the model methanol into the droplets with varying temperature and pressure. The x axis represents the inverse diffusion coefficient 1=Dg of the trace gas methanol. The square, circular, and triangular points denote T ¼ 13, 0, and 15 °C, respectively, and the black and white symbols denote He and Ar buffer gases. The solid line depicts the ideal deposition rate 1=Rideal of Eq. (7), and the dotted line depicts the linear regression line obtained via the method of least squares fit.
relation. The calculated data of 1=R show systematic upward deviation from 1=Rideal , implying augmented resistance in the gas-phase transport. This augmented resistance can be understood from the above-mentioned interference among droplets, which impedes the diffusional transport along the cylinder axis. These data are fairly well represented by a linear regression (dotted) line, but both the slope and the y intercept deviate considerably from the ideal (solid) line. Another noticeable feature observed in Fig. 3 is that data obtained at different temperatures are well scaled in the 1=Dg 1=R plot, indicating that the gas-phase diffusion coefficient Dg accounts for the influence of temperature on the deposition rate R. Subsequently, we calculated the uptake coefficient c of Eq. (2) with the finite surface deposition rate kint ¼ c=2, and tried to express the calculated 1=c with the resistance model formula Eq. (2), where the solubility resistance 1=Csol is neglected (1=Csol ¼ 0). Although we employed the full Fuchs–Sutugin formula of Eq. (3) for gas-phase resistance 1=Cg , the continuum expression (right limiting formula of Eqs. (3) or (4)) yields almost identical results within the region of interest. Fig. 4 shows the uptake resistance 1=c as a function of 1=Dg at T ¼ 0 °C. These data are well represented by Eq. (2) with the effective diameter deff and effective mass accommodation coefficient aeff , which respectively account for the slope and the y intercept in the 1=Dg 1=c plot. The optimized deff and aeff were determined to be 137ð 3Þ lm and 0.35( 0:03), respectively, via the method of least squares fit, where the error bars indicate the fitting accuracy. The effective droplet diameter deff ¼ 137 lm is significantly larger than the real diameter d ¼ 113 lm, by 20%. The concept of effective diameter has been utilized in experimental analysis in order to take into account gas-phase resistance [22], where the effective diameter is assumed to be twice the orifice diameter, 2do ¼ 120 lm. The present calculations reveal that the optimum diameter for the gas-phase resistance should be slightly larger than 2do . The effective mass accommodation coefficient aeff , which is apparent in the y intercept, is a controversial concept. Since we assumed twice the thermal collision rate for surface mass accommo-
M. Sugiyama et al. / Chemical Physics Letters 362 (2002) 56–62
61
sulfuric acid or organic liquids [22]. Otherwise, when the diffusive regime of low Knudsen numbers is necessarily employed as a consequence of vapor pressure being substantial, more accurate evaluation of the gas-phase resistance than that provided by the Fuchs–Sutugin formula Eq. (3) is desirable, in order to experimentally derive the mass accommodation coefficient.
4. Concluding remarks
Fig. 4. Inverse uptake coefficient 1=c of the model methanol as a function of inverse diffusion coefficient, 1=Dg , at T ¼ 0 °C. The black and white circles denote the He and Ar buffer gases, respectively, with varying pressure. The solid line depicts the analytical expression of Eqs. (2) and (3) with the optimized deff ¼ 137 lm and aeff ¼ 0:35, and the dashed line depicts the analytical expression of Eqs. (2) and (3) with the optimized deff ¼ 159 lm and a ¼ 1 fixed.
dation, this boundary condition should correspond to the unit mass accommodation coefficient a ¼ 1 [20]. Accordingly, we attempted a least squares fit of the data points in Fig. 4 with a fixed at 1, and the results are shown by the dashed line. However, the root mean square deviation of the dashed line is 3.2 times that of the solid line, indicating that, compared with the dashed line, the solid line is a more accurate representation. The dashed line implies that effective diameter is 159 lm, which is substantially larger than both the effective diameter of the solid line, 137 lm, and the real diameter, 113 lm. The above discussion suggests some uncertainty in the mass accommodation coefficient measured by the droplet train apparatus; interference among droplets tends to lead to underestimation of the mass accommodation coefficient. We stress that the most significant limitation of the above discussion lies in its being fully based on the continuum description of gas-phase diffusion. Therefore, the uncertainty of a suggested by the present discussion could be eliminated if a condition of sufficiently large Knudsen number (usually low pressure condition), where the gas-phase resistance is asymptotically negligible, is attained by employment of
The droplet train apparatus is one of the most powerful experimental means for measuring the mass accommodation coefficients a, due to provision of well-controlled surface area and short interaction time. However, the current experimental analysis for deriving a has some room for improvement, especially in the evaluation of the gasphase resistance. The gas-phase resistance has usually been estimated under an ideal assumption of static spherical boundary conditions around a droplet, whereas in the actual droplet tube, a train of droplets should be treated as being in flow. Therefore, in the present Letter we examined the validity of the above-mentioned ideal assumption and discussed accurate evaluation of the gas-phase resistance in the flow tube. We calculated the gasphase resistance directly, by numerically solving the coupled diffusion equation and fluid dynamics under boundary conditions that correspond to the experimental droplet flow tube. The calculated deposition rates reveal systematic deviation from the ideal model estimate based on the static spherical boundary condition, indicating augmented gas-phase resistance in the flow tube. This augmented resistance is attributed to interference among the droplets, which tends to develop cylindrical concentration field around the axis rather than the spherical distribution around each droplet. The calculated gas-phase resistance was well accounted for by the effective diameter deff and the effective mass accommodation coefficient aeff . The optimized effective diameter deff is significantly larger than the real droplet diameter d, by 20%, and also somewhat larger than twice the orifice diameter 2do . The effective mass accommodation coefficient aeff suggests some un-
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certainty in the experimentally derived value of a, which results from inaccurate estimation of gasphase resistance. This uncertainty could be significant at relatively low Knudsen numbers and large mass accommodation coefficients; e.g. in the case of water in water, and in such cases we suggest that experimental analysis requires more accurate treatment of gas-phase resistance. The present Letter reports the result of simulations for a flow tube reactor of typical geometry and conditions, and further quantitative comparison to the experiments should be performed under conditions that duplicate as closely as possible those employed in the experiments. A possible source of uncertainty of comparison with the experimental results lies in the calculated diffusion coefficients. From the simulation side, however, we note that the present results are robust to the diffusion coefficients employed. Whereas each data point of Figs. 3 or 4 is influenced by the value of the diffusion coefficient, the envelope curves, which express the diffusion-limited uptake rate as a function of diffusion coefficient, are confirmed to be quite insensitive. More comprehensive simulations conducted under varying conditions are in progress, towards determining general criteria useful in quantitatively evaluating gas-phase resistance under flow conditions. Acknowledgements The authors thank Drs. Doug Worsnop, Paul Davidovits and Mike Gershenson for their stimulating discussion and useful comments. This work was supported by Grants-in-Aid from the Ministry of Education and Science, Japan, and from AGS (Alliance for Global Sustainability) Project ‘Regional Climate and Air Quality.’
References [1] B.J. Finlayson-Pitts, J.N. Pitts Jr., Chemistry of the Upper and Lower Atmosphere, Academic Press, San Diego, 2000. [2] H.R. Pruppacher, J.D. Klett, Microphysics of Clouds and Precipitation, Kluwer Academic, Dordrecht, 1997. [3] J.H. Seinfeld, S.N. Pandis, Atmospheric Chemistry and Physics, Wiley, New York, 1998. [4] C.E. Kolb, D.R. Worsnop, M.S. Zahniser, P. Davidovits, L.F. Keyser, M.T. Leu, M.J. Molina, D.R. Hanson, A.R. Ravishankara, in: Progress and Problems in Atmospheric Chemistry, World Scientific, Singapore, 1995, p. 771. [5] S.E. Schwartz, in: Chemistry of Multiphase Atmospheric Systems, NATO ASI series, vol. G6, Springer, Berlin, 1986, p. 415. [6] D.R. Worsnop, M.S. Zahniser, C.E. Kolb, J.A. Gardner, L.R. Watson, J.M. Van Doren, J.T. Jayne, P. Davidovits, J. Phys. Chem. 93 (1989) 1159. [7] J.T. Jayne, S.X. Duan, P. Davidovits, D.R. Worsnop, M.S. Zahniser, C.E. Kolb, J. Phys. Chem. 95 (1991) 6329. [8] P. Davidovits, J.H. Hu, D.R. Worsnop, M.S. Zahniser, C.E. Kolb, Faraday Discuss. 100 (1995) 65. [9] Y.Q. Li, P. Davidovits, Q. Shi, J.T. Jayne, C.E. Kolb, D.R. Worsnop, J. Phys. Chem. A 105 (2001) 10627. [10] Y. Katrib, G. Deiber, F. Schweitzer, P. Mirabel, C. George, J. Aerosol Sci. 32 (2001) 893. [11] N.A. Fuchs, A.G. Sutugin, Highly Dispersed Aerosols, Ann Arbor Science Publishers, Ann Arbor, 1970. [12] D.R. Hanson, A.R. Ravishankara, E.R. Lovejoy, J. Geophys. Res. 101 (1996) 9063. [13] H. Motz, H. Wise, J. Chem. Phys. 32 (1960) 1893. [14] J.F. Widmann, E.J. Davis, J. Aerosol Sci. 28 (1997) 1233. [15] D.R. Lide (Ed.), Handbook of Chemistry and Physics, 81st ed., CRC Press, Boca Raton, 2000. [16] J.O. Hirschfelder, C.F. Curtiss, R.B. Bird, Molecular Theory of Gases and Liquids, Wiley, New York, 1954. [17] D.A. McQuarrie, Statistical Mechanics, Univ. Sci. Books, Sausalito, CA, 2000. [18] R.C. Reid, T.K. Sherwood, The Properties of Gases and Liquids, second ed., McGraw-Hill, New York, 1966. [19] M.V. Smoluchowski, Z. Phys. Chem. 92 (1918) 129. [20] K. Ibuki, M. Ueno, Bull. Chem. Soc. Jpn. 70 (1997) 543. [21] FLUENT 5.5, Fluent, inc., Lebanon, NH, 2000. [22] D.R. Worsnop, Q. Shi, J.T. Jayne, C.E. Kolb, E. Swartz, P. Davidovits, J. Aerosol Sci. 32 (2001) 877.
Chemical Physics Letters 362 (2002) 56–62 www.elsevier.com/locate/cplett
Numerical analysis of gas-phase diffusion resistance in a droplet train apparatus Masakazu Sugiyama
a,1
, Seiichiro Koda a, Akihiro Morita
b,*
a
b
Department of Chemical System Engineering, School of Engineering, The University of Tokyo, 7-3-1 Hongo, Bunkyo-ku, Tokyo 113-8656, Japan Department of Chemistry, Graduate School of Science, Kyoto University, Kitashirakawa, Sakyo-ku, Kyoto 606-8502, Japan Received 4 May 2002; in final form 6 June 2002
Abstract In this Letter, we theoretically analyze gaseous flow in a droplet train apparatus, which has been extensively utilized to study mass accommodation processes through gas/liquid interfaces. We numerically evaluate gas-phase resistance for uptake of trace gas into the droplets, via the coupled diffusion equation and fluid dynamics. In the results, gaseous resistance shows systematic deviation from values estimated from the ideal model with static spherical boundary conditions, indicating generation of additional resistance by interference among the train of droplets. Improved description of gas-phase resistance with optimized parameters in the flow tube is provided. Ó 2002 Elsevier Science B.V. All rights reserved.
1. Introduction Gas/liquid interfacial mass exchange is of critical significance in heterogeneous atmospheric chemistry and cloud microphysics [1–3]. The microscopic probability for interfacial mass transfer is characterized by the mass accommodation coefficient a, a ¼ðnumber of gas molecules absorbed into the liquidÞ =ðnumber of gas molecules impinging on the surfaceÞ;
ð1Þ *
Corresponding author. Fax: +81-75-753-4000. E-mail address: [email protected] (A. Morita). 1 Present address: Department of Electronics Engineering, School of Engineering, The University of Tokyo, 7-3-1 Hongo, Bunkyo-ku, Tokyo 113-8656, Japan.
which ranges from 0 to 1. However, experimentally determining the mass accommodation coefficient a from the phenomenological mass transfer rates [3,4] is not straightforward, since in most cases the interfacial mass exchange rates, characterized by uptake coefficients c, are influenced by numerous other factors, including gas-phase diffusion, solubility in the liquid phase, and reaction rate in/on the liquid. When chemical reactions in/ on the liquid are neglected, overall resistance of mass transfer 1=c is well described by the following decoupled form [4,5], 1 1 1 1 ¼ þ þ ; c Cg a Csol
ð2Þ
where the notations are as described in [4]; 1=Cg represents resistance of the gas-phase diffusion,
0009-2614/02/$ - see front matter Ó 2002 Elsevier Science B.V. All rights reserved. PII: S 0 0 0 9 - 2 6 1 4 ( 0 2 ) 0 1 0 1 9 - 9
M. Sugiyama et al. / Chemical Physics Letters 362 (2002) 56–62
and 1=Csol and represents the resistance of liquidphase solubility. Therefore, except for conditions where 1=Cg and 1=Csol are negligible, precise estimation of 1=Cg and 1=Csol is critical for experimental derivation of a. In the present Letter, we discuss accurate evaluation of the gas-phase resistance 1=Cg in a droplet train apparatus, since current methods of estimating 1=Cg might involve some uncertainty, as discussed below. The droplet train apparatus is one of the most widely employed experimental means for determining the mass accommodation coefficients [6– 10]. It utilizes a highly controlled train of droplets passing through a flow tube, and the uptake rate of the gaseous species into the droplets is determined from the reduction in the gaseous trace gas concentration after a certain gas–liquid interaction time. The droplets are generated by an orifice vibrating at a certain frequency and back pressure. The liquid surface is therefore continuously refreshed, and total surface area of the droplets is well controlled and varied by changing the orifice frequency. This apparatus utilizes relatively short and variable gas–liquid interaction time on the order of metersecond, thereby reducing the uncertainty in the solubility resistance 1=Csol . Droplet train experiments are usually conducted under low pressure conditions so as to minimize the gasphase resistance 1=Cg . These design features render this technique applicable within a wide range of the mass accommodation coefficients, including relatively large values up to unity. In the experimental analysis, the gas-phase resistance 1=Cg is usually evaluated via the Fuchs– Sutugin formula [11,12] or other alternative formulas [3]. The Fuchs–Sutugin formula describes the gas-phase resistance of uptake into a single spherical particle over a wide range of Knudsen number Kn, as follows 1 0:75 þ 0:283Kn ¼ Cg Knð1 þ KnÞ 0:75 0:467 ðfor low KnÞ; ð3Þ ! Kn where Kn ¼ 6Dg = cd; c is the average thermal vepffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi locity of the trace gas molecules ð¼ 8kB T =pM Þ; d the droplet diameter; and Dg the diffusion coefficient of the trace gas molecules. The limiting
57
formula of Eq. (3) in the continuum regime corresponds to the diffusive form of gas-phase resistance with the kinetic correction term 1/2 [4,13], 1 1 cd ¼ : Cg 8Dg 2
ð4Þ
The Fuchs–Sutugin formula Eqs. (3) or (4) involves an important assumption of the static spherical boundary condition around the droplet, whereas in an actual droplet train flow tube, a train of the droplets under flow should be considered [14]. Therefore, the purpose of this Letter is to examine the quantitative validity of Eq. (3) in droplet train flow tubes. We numerically solved the gaseous diffusion equation with fluid dynamics under flow conditions that correspond to the droplet train tube to thereby directly evaluate the gas-phase resistance. Accordingly, the present discussion focuses on the diffusive, continuum regime, which is often encountered under atmospheric conditions. The remainder of this Letter is constructed as follows. The next section explains the method and conditions employed for the calculations, and Section 3 provides results and discussion. Brief concluding remarks and future perspective follow in Section 4.
2. Calculation method and conditions Fig. 1 depicts the geometry of the boundary conditions. In a cylindrical flow tube having a diameter of 15 mm and a length of 20 mm, 50 spherical droplets are aligned on the axis at equal intervals. The droplet configuration is based on a typical set of experimental conditions; i.e. droplet velocity vd ¼ 27 m/s, orifice frequency fo ¼ 100 kHz, and orifice diameter do ¼ 60 lm [6]. Consequently, the center-of-mass distance between adjacent droplets is vd =fo ¼ 270 lm. The orifice diameter do ¼ 60 lm leads to the droplet diameter 1=3 d ¼ ð3do2 vd =2fo Þ 1:89do ¼ 113 lm via the liquid volume flow rate, F ¼
4p 3
3 2 d do fo ¼ p vd ; 2 2
ð5Þ
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M. Sugiyama et al. / Chemical Physics Letters 362 (2002) 56–62
Fig. 1. Cross-section of the cylindrical flow tube. Unit: mm. The wall and the gaseous flow move to the right in relation to the droplets.
and the Rayleigh natural breakup condition [6], fo ¼ vd =4:5do :
ð6Þ
The relative movement of the droplets to the wall was considered in actual calculations for fixed droplets. In Fig. 1, the wall of the flow tube is depicted as moving from left to right at a relative speed of vd with respect to the droplets. Calculations were performed under isothermal conditions within a temperature of )13–15 °C. Note that the droplets are supercooled but not frozen below 0 °C [6]. The gaseous volume flow rate is 200 cm3 =s, which means the gaseous average velocity is vg ¼ 200 cm3 =s=ðpð0:75 cmÞ2 Þ ¼ 1:13 m/ s. In order to incorporate this gaseous flow rate into the above boundary condition of fixed droplets and the moving tube wall, the average gaseous velocity at the inlet was set to vd vg for the same direction as the tube wall velocity. We assumed the sticking boundary condition at the surfaces of the wall and the droplets for the gaseous flow. The gas-phase species consists of a trace amount of methanol, saturated water vapor, and inert carrier gas of He or Ar. Initial methanol concentration is fixed as low as ng ¼ 1013 3 molecules=cm . For comparison, we also examined another concentration, 5 1013 molecules= cm3 , and confirmed the linearity between uptake rate and concentration. The saturated water vapor pressures are 1.7, 4.6, and 12.8 Torr at )13, 0, and
15 °C, respectively [4,15]. Total pressure ranges from 6 to 50 Torr at each temperature, with varying partial pressure of the carrier gas. The viscosity of the gas and the diffusion coefficients of the constituent molecules were evaluated by the Chapman–Enskog theory [16,17], where the collision integrals were calculated by the Stockmeyer model for polar molecules and by the Lennard– Jones model for nonpolar molecules. The ingredient parameters for the collision integrals were taken from literature [18]. This pressure range and the droplet diameter d assure the continuum description of the gas-phase diffusion, as the Knudsen number Kn K 0:2. In view of the low Reynolds numbers for the droplets, 0-5-10, we can safely assume laminar gaseous flow in the flow tube. Regarding the intrinsic deposition rate kint of the trace gas onto the droplet surface, we assumed two conditions-perfect uptake, and a finite rate kint ¼ c=2. The former condition of perfect uptake was actually implemented with a sufficiently large rate, kint ¼ 1035 cm=s 2, which effectively imposed vanishing concentration of the trace gas at the droplet surface. This is consistent with the usual Smoluchowski boundary condition [19]. Whereas the former condition completely neglects the intrinsic resistance of the mass accommodation, the 2
We compared another value kint ¼ 1035 cm=s to 1030 , and confirmed derivation of identical results, indicating that the present kint is sufficiently large.
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latter condition effectively takes into account the finite collision rate [20]. The finite rate kint ¼ c=2 is twice the thermal collision rate c=4, corresponding to the kinetic correction term 1/2 in Eq. (4). Although the two assumptions yield little difference in the overall resistance in the continuum regime, we employed the latter assumption in order to discuss effective diameter and mass accommodation coefficient later. Numerical calculation of the coupled fluid and diffusion equations was performed by use of the F L U E N T package [21] in twodimensional cylindrical space. A total of 64,413 triangle mesh points were employed, including particularly fine radial mesh points near the gas/ droplet interface, which is the critical region for the uptake. To check the accuracy yielded by the mesh size, we used another finer mesh geometry, consisting of 110,626 quadrilateral grids, and confirmed that the calculated deposition rates for the two cell geometries agree within 0.5%.
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3. Results and discussion Fig. 2 depicts steady-state gaseous velocity and trace gas concentration fields in the flow tube. The velocity field in Fig. 2a is nearly uniform except for the vicinity of the axis, where the strong perturbation is apparent, owing to the train of droplets. Relative gaseous velocity on the axis between the droplets is quite small, implying that the volume elements of the gas lying between the droplets pass along the axis at nearly the same speed as the droplets. This is consistent with experimental evidence that the train of the droplets suffers from little deceleration by frictional drag in the flow tube [22]. The concentration field of the trace gas in Fig. 2b shows substantial depletion in the vicinity of the droplet, train, and on the whole, cylindrical concentration distribution around the axis is developed rather than spherical distribution around each droplet. This feature of the emphasized radial
Fig. 2. Steady-state (a) gaseous velocity field and (b) concentration field of methanol in the flow tube at 0 °C and 25 Torr. Half cross sections of the cylinder tube are shown, where the top and bottom lines correspond to the wall and the axis, respectively. In panel (a), relative velocity of the cylinder wall with respect to the droplets is 27 m/s. In panel (b), contour spacing is ng =20. A magnified figure in the vicinity of the droplets is inserted in the upper right corner of panel (b).
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concentration gradient is attributed to the axial flow and to interference among the train of droplets. Since the axial concentration gradient becomes relatively unimportant with respect to the gaseous diffusion, the gas-phase diffusional transport approaches two-dimensional transport, implying deviation from Eqs. (3) or (4). The deposition rate of the trace gas methanol into the droplets per unit surface area, R, was calculated by the diffusion equation with the absorbing boundary at the droplet surface and compared with the ideal, spherical case. In the latter, ideal case, the steady-state Smoluchowski equation around a single spherical droplet of the diameter d yields a deposition rate Rideal ¼ 2Dg ng =d, which corresponds to the continuum regime of Eq. (4) without the kinetic correction term 1/2 and the normalization factor ng c=4; !, ! 1 d ng c cd ¼ ¼ : ð7Þ Rideal 2Dg ng 8Dg 4 Fig. 3 compares the calculated deposition rates R and the ideal deposition rates Rideal in the 1=Dg 1=R plot, where the latter should describe a linear
Fig. 3. Inverse deposition rate 1=R of the model methanol into the droplets with varying temperature and pressure. The x axis represents the inverse diffusion coefficient 1=Dg of the trace gas methanol. The square, circular, and triangular points denote T ¼ 13, 0, and 15 °C, respectively, and the black and white symbols denote He and Ar buffer gases. The solid line depicts the ideal deposition rate 1=Rideal of Eq. (7), and the dotted line depicts the linear regression line obtained via the method of least squares fit.
relation. The calculated data of 1=R show systematic upward deviation from 1=Rideal , implying augmented resistance in the gas-phase transport. This augmented resistance can be understood from the above-mentioned interference among droplets, which impedes the diffusional transport along the cylinder axis. These data are fairly well represented by a linear regression (dotted) line, but both the slope and the y intercept deviate considerably from the ideal (solid) line. Another noticeable feature observed in Fig. 3 is that data obtained at different temperatures are well scaled in the 1=Dg 1=R plot, indicating that the gas-phase diffusion coefficient Dg accounts for the influence of temperature on the deposition rate R. Subsequently, we calculated the uptake coefficient c of Eq. (2) with the finite surface deposition rate kint ¼ c=2, and tried to express the calculated 1=c with the resistance model formula Eq. (2), where the solubility resistance 1=Csol is neglected (1=Csol ¼ 0). Although we employed the full Fuchs–Sutugin formula of Eq. (3) for gas-phase resistance 1=Cg , the continuum expression (right limiting formula of Eqs. (3) or (4)) yields almost identical results within the region of interest. Fig. 4 shows the uptake resistance 1=c as a function of 1=Dg at T ¼ 0 °C. These data are well represented by Eq. (2) with the effective diameter deff and effective mass accommodation coefficient aeff , which respectively account for the slope and the y intercept in the 1=Dg 1=c plot. The optimized deff and aeff were determined to be 137ð 3Þ lm and 0.35( 0:03), respectively, via the method of least squares fit, where the error bars indicate the fitting accuracy. The effective droplet diameter deff ¼ 137 lm is significantly larger than the real diameter d ¼ 113 lm, by 20%. The concept of effective diameter has been utilized in experimental analysis in order to take into account gas-phase resistance [22], where the effective diameter is assumed to be twice the orifice diameter, 2do ¼ 120 lm. The present calculations reveal that the optimum diameter for the gas-phase resistance should be slightly larger than 2do . The effective mass accommodation coefficient aeff , which is apparent in the y intercept, is a controversial concept. Since we assumed twice the thermal collision rate for surface mass accommo-
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sulfuric acid or organic liquids [22]. Otherwise, when the diffusive regime of low Knudsen numbers is necessarily employed as a consequence of vapor pressure being substantial, more accurate evaluation of the gas-phase resistance than that provided by the Fuchs–Sutugin formula Eq. (3) is desirable, in order to experimentally derive the mass accommodation coefficient.
4. Concluding remarks
Fig. 4. Inverse uptake coefficient 1=c of the model methanol as a function of inverse diffusion coefficient, 1=Dg , at T ¼ 0 °C. The black and white circles denote the He and Ar buffer gases, respectively, with varying pressure. The solid line depicts the analytical expression of Eqs. (2) and (3) with the optimized deff ¼ 137 lm and aeff ¼ 0:35, and the dashed line depicts the analytical expression of Eqs. (2) and (3) with the optimized deff ¼ 159 lm and a ¼ 1 fixed.
dation, this boundary condition should correspond to the unit mass accommodation coefficient a ¼ 1 [20]. Accordingly, we attempted a least squares fit of the data points in Fig. 4 with a fixed at 1, and the results are shown by the dashed line. However, the root mean square deviation of the dashed line is 3.2 times that of the solid line, indicating that, compared with the dashed line, the solid line is a more accurate representation. The dashed line implies that effective diameter is 159 lm, which is substantially larger than both the effective diameter of the solid line, 137 lm, and the real diameter, 113 lm. The above discussion suggests some uncertainty in the mass accommodation coefficient measured by the droplet train apparatus; interference among droplets tends to lead to underestimation of the mass accommodation coefficient. We stress that the most significant limitation of the above discussion lies in its being fully based on the continuum description of gas-phase diffusion. Therefore, the uncertainty of a suggested by the present discussion could be eliminated if a condition of sufficiently large Knudsen number (usually low pressure condition), where the gas-phase resistance is asymptotically negligible, is attained by employment of
The droplet train apparatus is one of the most powerful experimental means for measuring the mass accommodation coefficients a, due to provision of well-controlled surface area and short interaction time. However, the current experimental analysis for deriving a has some room for improvement, especially in the evaluation of the gasphase resistance. The gas-phase resistance has usually been estimated under an ideal assumption of static spherical boundary conditions around a droplet, whereas in the actual droplet tube, a train of droplets should be treated as being in flow. Therefore, in the present Letter we examined the validity of the above-mentioned ideal assumption and discussed accurate evaluation of the gas-phase resistance in the flow tube. We calculated the gasphase resistance directly, by numerically solving the coupled diffusion equation and fluid dynamics under boundary conditions that correspond to the experimental droplet flow tube. The calculated deposition rates reveal systematic deviation from the ideal model estimate based on the static spherical boundary condition, indicating augmented gas-phase resistance in the flow tube. This augmented resistance is attributed to interference among the droplets, which tends to develop cylindrical concentration field around the axis rather than the spherical distribution around each droplet. The calculated gas-phase resistance was well accounted for by the effective diameter deff and the effective mass accommodation coefficient aeff . The optimized effective diameter deff is significantly larger than the real droplet diameter d, by 20%, and also somewhat larger than twice the orifice diameter 2do . The effective mass accommodation coefficient aeff suggests some un-
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certainty in the experimentally derived value of a, which results from inaccurate estimation of gasphase resistance. This uncertainty could be significant at relatively low Knudsen numbers and large mass accommodation coefficients; e.g. in the case of water in water, and in such cases we suggest that experimental analysis requires more accurate treatment of gas-phase resistance. The present Letter reports the result of simulations for a flow tube reactor of typical geometry and conditions, and further quantitative comparison to the experiments should be performed under conditions that duplicate as closely as possible those employed in the experiments. A possible source of uncertainty of comparison with the experimental results lies in the calculated diffusion coefficients. From the simulation side, however, we note that the present results are robust to the diffusion coefficients employed. Whereas each data point of Figs. 3 or 4 is influenced by the value of the diffusion coefficient, the envelope curves, which express the diffusion-limited uptake rate as a function of diffusion coefficient, are confirmed to be quite insensitive. More comprehensive simulations conducted under varying conditions are in progress, towards determining general criteria useful in quantitatively evaluating gas-phase resistance under flow conditions. Acknowledgements The authors thank Drs. Doug Worsnop, Paul Davidovits and Mike Gershenson for their stimulating discussion and useful comments. This work was supported by Grants-in-Aid from the Ministry of Education and Science, Japan, and from AGS (Alliance for Global Sustainability) Project ‘Regional Climate and Air Quality.’
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