B. Krasovitov, T. Elperin and A. Fominykh Department of Mechanical Engineering

1. Evaporation and condensation of droplets in the presence of inert admixtures containing soluble gas B. Krasovitov, T. Elperin and A. Fominykh 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

2. Outline of the presentation • Motivation and goals • Fundamentals • Description of the model • Results and discussion • Conclusions Ben-Gurion University of the Negev

3. Gas absorption by droplets Air Soluble gas Scavenging of air pollutions Spray tower absorbers • SO2 absorption of boiler flue gas • HF absorption in the aluminum industry • In-cloud scavenging of polluted gases (SO2, CO2, CO, NOx, NH3) Spray scrubbers Single Droplet Ben-Gurion University of the Negev

4. Gas absorption by stagnant droplets: Scientific background • Dispersed-phase controlled isothermal absorption model (Newman A. B., 1931) Particle Sherwood number: where Ben-Gurion University of the Negev

5. Gas absorption by stagnant droplets: Scientific background • Gas absorption in the presence of inert admixtures (see e.g., Plocker U.J., Schmidt-Traub H., 1972) • 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 soluble gas (see e.g., Ray A. K., Huckaby J. L. and Shah T., 1987, 1989); • 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). Ben-Gurion University of the Negev

6. Absorption equilibria Air SO2 is the species in dissolved state Henry’s Law: Aqueous phase sulfur dioxide/water chemical equilibria Droplet Gas-liquid interface = molecule of soluble gas = pollutant captured in solution Total dissolved sulfur in solution in oxidation state 4: Ben-Gurion University of the Negev

7. Absorption equilibria: Aqueous phase sulfur dioxide/water chemical equilibria The equilibrium constants (Maahs, 1982): (1) Langmuir’s Electro neutrality principle (1920): Electro neutrality equation: Huckaby & Ray (1989), Walcek et al. (1984): (2) Ben-Gurion University of the Negev

8. Absorption equilibria: Aqueous phase sulfur dioxide/water chemical equilibria Eqs. (1) – (2) yield the following equation for concentration of ions : pHis a measure of theacidity oralkalinityof a solution. (3) Using (1) – (3) we obtain: (4) Eq. (4) yields the following expression for the effective Henry's constant: Figure 1. Equilibrium dissolved S(IV) as a function of pH, gas-phase partial pressure of SO2 and pressure (Seinfeld, 1986). (5) Ben-Gurion University of the Negev

9. Gas absorption by stagnant droplet: Description of the model Gaseous phase Z Gas-liquid interface q R Far field Droplet Y j X Governing equations 1. gaseous phaser > R (t) (6) (7) (8) 2. liquid phase0 < r < R (t) (9) (10) In Eqs. (7) Ben-Gurion University of the Negev

10. Gas absorption by stagnant droplet: Description of the model • anelastic approximation: • subsonic flow velocities (low Mach number approximation, M << 1) (11) (12) In spherical coordinates Eq. (11) reads: The radial flow velocity can be obtained by integrating equation (12): (13) (14) Ben-Gurion University of the Negev

11. Description of the model Stefan velocity and droplet vaporization rate The continuity condition for the radial flux of the absorbate at the droplet surface reads (Elperin et al. 2005, 2007): (15) Other non-soluble components of the inert admixtures are not absorbed in the liquid (16) Taking into account Eq. (16) and using anelastic approximation (Eq.12) we can obtain the expression for Stefan velocity: (17) where subscript “1” denotes water vapor species Ben-Gurion University of the Negev

12. Description of the model Stefan velocity and droplet vaporization rate The material balance at the gas-liquid interface yields: (18) Then assuming we obtain the following expressions for the rate of change of droplet's radius (Elperin et al. 2005, 2007): (19) Ben-Gurion University of the Negev

13. Description of the model 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 Stefan velocity and droplet vaporization rate Ben-Gurion University of the Negev

14. Description of the model Stefan velocity and droplet vaporization rate Huckaby and Ray (1989) Ben-Gurion University of the Negev

15. Description of the model Initial and boundary conditions The initial conditions for the system of equations (6)–(10) read: At t = 0, (20) At t = 0, At the droplet surface: (21) (22) (23) (24) Ben-Gurion University of the Negev

16. Description of the model Initial and boundary conditions The equilibrium between solvable gaseous and dissolved in liquid species reads: (25) where (26) Huckaby & Ray (1989): Ben-Gurion University of the Negev

17. Description of the model Vapor concentration at the droplet surface and Henry’s constant The vapor concentration (1-st species) at the droplet surface is the function of temperature Ts(t) and can be determined as follows: (27) The functional dependence of the Henry's law constant vs. temperature reads: (28) Figure 2. Henry’s constant vs. temperature Ben-Gurion University of the Negev

18. Description of the model Initial and boundary conditions In the center of the droplet symmetry conditions yields: (27) At and the ‘soft’ boundary conditions at infinity are imposed: (28) Ben-Gurion University of the Negev

19. Method of numerical solution • 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 (6)–(10) • was solved using the method of lines; • The mesh points are spaced adaptively using the following formula: Ben-Gurion University of the Negev

20. Results and discussion Average concentration of absorbed CO2 in the droplet: Analytical solution in the case of aqueous-phase controlled diffusion in a stagnant non-evaporating droplet: Figure 3. Comparison of the numerical results with the experimental data (Taniguchi & Asano, 1992) and analytical solution (Elperin et al 2005). Ben-Gurion University of the Negev

21. Results and discussion Average concentration of the absorbed SO2 in the droplet: relative absorbate concentration is determined as follows: Figure 4. Dependence of average aqueous sulfur dioxide molar concentration vs. time for various values of relative humidity (Elperin et al. 2005). Figure 5. Dependence of dimensionless average aqueous SO2 concentration vs. time for various initial sizes of evaporating droplet R0 (Elperin et al. 2005). Ben-Gurion University of the Negev

22. Results and discussion Figure 8. Droplet surface temperature vs. time: 1 – model taking into account the equilibrium dissociation reactions; 2 – model of physical absorption (Elperin et al., 2007). Figure 7. Effect of Stefan flow and heat of absorption on droplet surface temperature (Elperin et al. 2005). Figure 6. Droplet surface temperature vs. time (Elperin et al., 2007). Ben-Gurion University of the Negev

23. Results and discussion Figure 9b.Temporal evolution of surface temperature for a water droplet evaporating in N2/NH3/H2O gaseous mixture (Elperin et al., 2007). Figure 9a.Droplet surface temperature vs. time (Elperin et al., 2007). Ben-Gurion University of the Negev

24. Results and discussion Figure 10b.Droplet surface temperature vs. time, [SO2(g)]0 = 10-6 mole/m3 (Huckaby and Ray, 1989). Figure 10a.Droplet surface temperature vs. time, YA = 0.01 (Huckaby and Ray, 1987). Ben-Gurion University of the Negev

25. Results and discussion Figure 12.Temporal evolution of surface temperature for a water droplet evaporating in N2/SO2/H2O gaseous mixture (Elperin et al., 2005). Figure 11.Temporal evolution of the surface temperature for a water droplet condensation in N2/CO2/H2O gaseous mixture (Elperin et al., 2005). Ben-Gurion University of the Negev

26. Results and discussion Figure 14. Average concentration of aqueous sulfur species and their sum vs. time, RH = 101% (Elperin et al., 2007). Figure 13. Average concentration of aqueous sulfur species and their sum vs. time, RH = 70% (Elperin et al., 2007). Ben-Gurion University of the Negev

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

28. Conclusion 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 absorbate 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, and in design calculations of gas-liquid contactors. Ben-Gurion University of the Negev