T. Elperin, A. Fominykh and B. Krasovitov Department of Mechanical Engineering The Pearlstone Center for Aeronautical Engineering Studies Ben-Gurion University of the Negev P.O.B. 653, Beer Sheva 84105, ISRAEL

Motivation and goals Fundamentals Description of the model Results and discussion Conclusions Ben-Gurion University of the Negev Summer Heat Transfer Conference San Francisco, CA, July 19-23, 2009

Ben-Gurion University of the Negev Single Droplet Atmospheric polluted gases (SO 2, CO 2, CO, NOx, NH 3 ): Air Soluble gas Scavenging of air pollutions by cloud and rain droplets is the species in dissolved state Henry’s Law: In-cloud scavenging of polluted gases Scavenging of air pollutions by rain droplets Summer Heat Transfer Conference San Francisco, CA, July 19-23, 2009

Ben-Gurion University of the Negev Scavenging of air pollutions Gaseous pollutants in atmosphere SO 2 and NH 3 – anthropogenic emission CO 2 – competition between photosynthesis, respiration and thermally driven buoyant mixing Fig. 1a. Aircraft observation of vertical profiles of CO 2 concentration (by Perez-Landa et al., 2007) Summer Heat Transfer Conference San Francisco, CA, July 19-23, 2009

Ben-Gurion University of the Negev Scavenging of air pollutions Gaseous pollutants in atmosphere SO 2 and NH 3 – anthropogenic emission CO 2 – competition between photosynthesis, respiration and thermally driven buoyant mixing Summer Heat Transfer Conference San Francisco, CA, July 19-23, 2009 Fig. 1b. Vertical distribution of SO 2. Solid lines - results of calculations with (1) and without (2) wet chemical reaction (Gravenhorst et al. 1978); experimental values (dashed lines) – (a) Georgii & Jost (1964); (b) Jost (1974); (c) Gravenhorst (1975); Georgii (1970); Gravenhorst (1975); (f) Jaeschke et al., (1976)

Ben-Gurion University of the Negev Scavenging of air pollutions Vertical temperature profile in the lowest few kilometers of the atmosphere Adiabatic decrease of atmospheric temperature with height Inversion of vertical temperature gradient as a result of solar radiation heating and ground cooling Summer Heat Transfer Conference San Francisco, CA, July 19-23, 2009 Fig. 1c. Aircraft observation of potential temperature vs. height (by Perez-Landa et al., 2007)

Gas absorption by falling droplets: Walcek and Pruppacher, 1984 Alexandrova et al., 2004 Elperin and Fominykh, 2005 Measurements of vertical distribution of trace gases in the atmosphere: SO 2 – Gravenhorst et al., 1978 NH 3 – Georgii & Müller, 1974 CO 2 – Denning et al., 1995; Perez-Landa et al., 2007 Scavenging of gaseous pollutants by falling rain droplets in inhomogeneous atmosphere: Elperin, Fominykh & Krasovitov 2008 – non-uniform concentration distribution in a gaseous phase Elperin, Fominykh & Krasovitov 2009 – non-uniform temperature and concentration distribution in the atmosphere Ben-Gurion University of the Negev Summer Heat Transfer Conference San Francisco, CA, July 19-23, 2009

Ben-Gurion University of the Negev In the analysis we used the following assumptions:  c << R;  T << R Tangential molecular mass transfer rate along the surface is small compared with a molecular mass transfer rate in the normal direction The bulk of a droplet, beyond the diffusion and temperature boundary layers, is completely mixed by circulations inside a droplet The droplet has a spherical shape. Fig. 2. Schematic view of a falling droplet and temperature and concentration profiles 0.1 mm R 0.5 mm 10 Re U 4.5 m/s Summer Heat Transfer Conference San Francisco, CA, July 19-23, 2009

Ben-Gurion University of the Negev Fluid velocity components at the gas-liquid interface are (Prippacher & Klett, 1997): (1) System of convective diffusion and energy conservation transient equations for the liquid and gaseous phases read: (2) (i = 1, 2) where k = for different Re, and Summer Heat Transfer Conference San Francisco, CA, July 19-23, 2009

Ben-Gurion University of the Negev Initial and boundary conditions where Summer Heat Transfer Conference San Francisco, CA, July 19-23, 2009 Using the transformation z = U t the coordinate-dependent boundary conditions can be transformed into the time-dependent boundary conditions: (3) at (4) (5) (6) (7) at (8) (9) (10)

Ben-Gurion University of the Negev (11) where Introduction of the self-similar variables: and application of Duhamel's theorem yields a solution of convection diffusion and energy conservation equations (Eqs. 1): and Summer Heat Transfer Conference San Francisco, CA, July 19-23, 2009

Ben-Gurion University of the Negev Integral energy and material balances over the droplet yields: (12) Substituting solutions (11) into Eqs. (12) yields: where Summer Heat Transfer Conference San Francisco, CA, July 19-23, 2009 (13) (14)  initial value of molar fraction of absorbate in a droplet  value of molar fraction of an absorbate in a gas phase at height H

Ben-Gurion University of the Negev The system of equations for temperature and absorbate concentration in the bulk of a droplet: is a system of linear convolution Volterra integral equations of the second kind that can be written in the following form: (15) Summer Heat Transfer Conference San Francisco, CA, July 19-23, 2009 is a system of linear convolution Volterra integral equations of the second kind that can be written in the following form: (15)

The method of solution is based on the approximate calculation of a definite integral using some quadrature formula: The uniform mesh with an increment h was used: Using trapezoidal integration rule we obtain a system of linear algebraic equations: The kernel is a matrix and equation (17) is viewed as a vector equation. Ben-Gurion University of the Negev where – remainder of the series after the N-th term. (16) (17) Summer Heat Transfer Conference San Francisco, CA, July 19-23, 2009

Ben-Gurion University of the Negev. Fig. 3. Dependence of CO2 concentration in the atmosphere vs. altitude (1) aircraft measurements Valencia 6:23 (by Perez-Landa et al., 2007); (2)-(4) approximation of the measured data; (3) aircraft measurements Valencia 13:03 (by Perez-Landa et al., 2007). Summer Heat Transfer Conference San Francisco, CA, July 19-23, 2009 Fig. 4. Dependence of the potential, atmospheric and droplet surface temperature vs. altitude in the morning.

Ben-Gurion University of the Negev. Fig. 3. Dependence of CO2 concentration in the atmosphere vs. altitude (1) aircraft measurements Valencia 6:23 (by Perez-Landa et al., 2007); (2)-(4) approximation of the measured data; (3) aircraft measurements Valencia 13:03 (by Perez-Landa et al., 2007). Summer Heat Transfer Conference San Francisco, CA, July 19-23, 2009 Fig. 5. Dependence of the potential, atmospheric and droplet surface temperature vs. altitude in the afternoon

Ben-Gurion University of the Negev. Fig. 3. Dependence of CO2 concentration in the atmosphere vs. altitude (1) aircraft measurements Valencia 6:23 (by Perez-Landa et al., 2007); (2)-(4) approximation of the measured data; (3) aircraft measurements Valencia 13:03 (by Perez-Landa et al., 2007). Summer Heat Transfer Conference San Francisco, CA, July 19-23, 2009 Fig. 6. Dependence of the concentration of the dissolved CO 2 gas in the bulk of a falling rain droplet vs. time, x b10 = 0.

Ben-Gurion University of the Negev Summer Heat Transfer Conference San Francisco, CA, July 19-23, 2009 Fig. 7. Dependence of the interfacial temperature of a falling rain droplet vs. altitude. Fig. 8. Dependence of the relative concentration of ammonia (NH 3 ) inside a water droplet vs. time.

Ben-Gurion University of the Negev Summer Heat Transfer Conference San Francisco, CA, July 19-23, 2009 Fig. 9. Evolution of ammonia (NH 3 ) distribution in the atmosphere due to scavenging by rain

Ben-Gurion University of the Negev The suggested model of gas absorption by a falling liquid droplet in the presence of inert admixtures takes into account a number of effects that were neglected in the previous studies, such as the effect of dissolved gas accumulation inside a droplet and effect of the absorbate and temperature inhomogenity in a gaseous phase on the rate of heat and mass transfer. It is shown than if concentration of a trace gas in the atmosphere is homogeneous and temperature in the atmosphere decreases with height, beginning from some altitude gas absorption is replaced by gas desorption. We found that the neglecting temperature inhomogenity in the atmosphere described by adiabatic lapse rate leads to overestimation of trace gas concentration in a droplet at the ground on tens of percents. If concentration of soluble trace gas is homogeneous and temperature increases with height e.g. during the nocturnal inversion, droplet absorbs gas during all the time of its fall. Summer Heat Transfer Conference San Francisco, CA, July 19-23, 2009