Numerical analysis of simultaneous heat and mass transfer during absorption of polluted gases by cloud droplets 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 Description of the model Results and discussion Conclusions Outline of the presentation Ben-Gurion University of the Negev

Effect of vapor condensation at the surface of stagnant droplets on the rate of mass transfer during gas absorption by growing droplets: uniform temperature distribution in both phases was assumed (see e.g., Karamchandani, P., Ray, A. K. and Das, N., 1984); liquid-phase controlled mass transfer during absorption was investigated when the system consisted of liquid droplet, its vapor and solvable gas (see e.g., Ray A. K., Huckaby J. L. and Shah T., 1987, 1989); Gas absorption by falling droplets accompanied by subsequent dissociation reaction (see e.g., Baboolal et al. (1981), Walcek and Pruppacher (1984), Alexandrova et al., 2004); Simultaneous heat and mass transfer during droplet evaporation or growth: model of physical absorption (Elperin et al., 2005); model taking into account subsequent dissociation reaction (Elperin et. al, 2007). Gas absorption by cloud droplets: Scientific background Ben-Gurion University of the Negev

Gas-liquid interface Droplet Absorption equilibria Ben-Gurion University of the Negev = pollutant molecule = pollutant captured in solution Air SO 2 Aqueous phase sulfur dioxide/water chemical equilibria is the species in dissolved state Henry’s Law: Electro neutrality equation:

Description of the model Ben-Gurion University of the Negev Governing equations 1. gaseous phase r > R (t) 2. liquid phase 0 < r < R (t) (1) (2) (3) (4) (5) In Eqs. (2) Droplet Far field  Z X Y  Gas- liquid interface R Gaseous phase

Description of the model Ben-Gurion University of the Negev The continuity condition for the radial flux of the absorbate at the droplet surface reads: Other non-soluble components of the inert admixtures are not absorbed in the liquid (6) (7) Taking into account Eq. (7) and using anelastic approximation we can obtain the expression for Stefan velocity: (8) where subscript “ 1 ” denotes water vapor species Stefan velocity and droplet vaporization rate

Description of the model Ben-Gurion University of the Negev The material balance at the gas-liquid interface yields: (9) Then assuming we obtain the following expressions for the rate of change of droplet's radius: (10) Stefan velocity and droplet vaporization rate

Description of the model Ben-Gurion University of the Negev Stefan velocity and droplet vaporization rate In the case when all of the inert admixtures are not absorbed in liquid the expressions for Stefan velocity and rate of change of droplet radius read

Description of the model Ben-Gurion University of the Negev Initial and boundary conditions The initial conditions for the system of equations (1)–(5) read: At t = 0, (11) At the droplet surface: (12) (13) (14) (15)

Description of the model Ben-Gurion University of the Negev Initial and boundary conditions The equilibrium between solvable gaseous and dissolved in liquid species can be expressed using the Henry's law (16) where (17) In the center of the droplet symmetry conditions yields: (18) (19) At and the ‘ soft ’ boundary conditions at infinity are imposed

Spatial coordinate transformation: The gas-liquid interface is located at Coordinates x and w can be treated identically in numerical calculations; Time variable transformation: The system of nonlinear parabolic partial differential equations (1)–(5) was solved using the method of lines; The mesh points are spaced adaptively using the following formula: Method of numerical solution Ben-Gurion University of the Negev

Results and discussion Ben-Gurion University of the Negev Average concentration of the absorbed SO 2 in the droplet: Figure 1. Dependence of average aqueous sulfur dioxide molar concentration vs. time for various values of relative humidity. Figure 2. Dependence of dimensionless average aqueous SO 2 concentration vs. time for various initial sizes of evaporating droplet R 0. relative absorbate concentration is determined as follows:

Results and discussion Ben-Gurion University of the Negev Figure 3. Droplet surface temperature vs. time (T 0 = 278 K, T ∞ = 293 K, RH = 70%). Figure 4. Effect of Stefan flow and heat of absorption on droplet surface temperature (Elperin et al. 2005). Figure 5. Droplet surface temperature vs. time: 1 – model taking into account the equilibrium dissociation reactions; 2 – model of physical absorption.

Results and discussion Ben-Gurion University of the Negev Figure 7. Average concentration of aqueous sulfur species and their sum vs. time ( RH = 101%). Figure 6. Average concentration of aqueous sulfur species and their sum vs. time ( RH = 70%). pH is a measure of the acidity or alkalinity of a solution.

Results and discussion: the interrelation between heat and mass transport Ben-Gurion University of the Negev Decreases Stefan velocity Decreases vapor flux Increases droplet surface temperature Increases vapor flux Decreases effective Henry’s constant Decreases droplet surface temperature Increases absorbate flux Increases effective Henry’s constant Decreases absorbate flux Increases Stefan velocity

Conclusion Ben-Gurion University of the Negev The obtained results show, that the heat and mass transfer rates in water droplet-air-water vapor system at short times are considerably enhanced under the effects of Stefan flow, heat of absorption and dissociation reactions within the droplet. It was shown that nonlinearity of the dependence of droplet surface temperature vs. time stems from the interaction of different phenomena. Numerical analysis showed that in the case of small concentrations of SO 2 in a gaseous phase the effects of Stefan flow and heat of absorption on the droplet surface temperature can be neglected. The developed model allows to calculate the value of pH vs. time for both evaporating and growing droplets. The performed calculations showed that the dependence of pH increase with the increasing relative humidity (RH). The performed analysis of gas absorption by liquid droplets accompanied by droplets evaporation and vapor condensation on the surface of liquid droplets can be used in calculations of scavenging of hazardous gases in atmosphere by rain, atmospheric cloud evolution.