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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 Outline of the presentation Motivation and goals Fundamentals Description of the model Results and discussion Conclusions Ben-Gurion University of the Negev Gas absorption by droplets Spray tower Scavenging of absorbers • SO2 absorption of air pollutions boiler flue gas • HF absorption in the aluminum industry • In-cloud scavenging of polluted gases (SO2, CO2, CO, NOx, NH3) Spray scrubbers Air Soluble gas Single Droplet Ben-Gurion University of the Negev Gas absorption by stagnant droplets: Scientific background Dispersed-phase controlled isothermal absorption model (Newman A. B., 1931) C t 6 1 n 2 π 2 Dl t 1 2 2 exp Csat π n 1 n R2 Particle Sherwood number: exp n 2 2 2 2 n 1 Sh p 3 1 2 exp n 2 2 n 1 n Dl t where R2 Ben-Gurion University of the Negev 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 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 Absorption equilibria: Aqueous phase sulfur dioxide/water chemical equilibria The equilibrium constants (Maahs, 1982): 1376.1 853 [ SO2 H 2O] 4.521 [H ][ HSO3 ] 4.74 KH 10 T ; K1 10 T pSO2 [ SO2 H 2O] (1) 6717 621.9 1 1 [ H ][ SO3 ] 2 9.278 K2 10 T ; Kw H 14 OH 1.008 10 e T 298 [ HSO3 ] Langmuir‟s Electro neutrality principle (1920): Ci zi 0 Electro neutrality equation: H HSO3 2SO32 Huckaby & Ray (1989), Walcek et al. (1984): (2) Ben-Gurion University of the Negev Absorption equilibria: Aqueous phase sulfur dioxide/water chemical equilibria Eqs. (1) – (2) yield the following equation for concentration of ions : (3) Using (1) – (3) we obtain: pH is a measure of the acidity or alkalinity of a solution . (4) Eq. (4) yields the following expression for the effective Henry's constant: Figure 1. Equilibrium dissolved (5) S(IV) as a function of pH, gas-phase partial pressure of SO2 and pressure (Seinfeld, 1986). Ben-Gurion University of the Negev Gas absorption by stagnant droplet: Description of the model Governing equations 1. gaseous phase r > R (t) r2 2 t r r vr 0 (6) Droplet Far field Gaseous phase 2 Y j r 2 t Y j vr r Y j D j r (7) r 2 r r Z d mL ds c pTe r 2 t r T vr r 2c pTe ke r 2 e (8) r r mA q Gas- R liquid interface 2. liquid phase 0 < r < R (t) Y T L 2 T L r 2 L r (9) j t r r X 2 YA 2 ( L) L r LYA L DL r (10) j 1,..., K 1, r In Eqs. (7) t r S IV M S (IV ) K L Y j 1 j 1; YA Ben-Gurion University of the Negev Gas absorption by stagnant droplet: Description of the model anelastic approximation: v 2 c 2 1 Eq. 6 v 0. (11) In spherical coordinates Eq. (11) reads: 2 r r vr 0 (12) The radial flow velocity can be obtained by integrating equation (12): vr r 2 const (13) subsonic flow velocities (low Mach number approximation, M << 1) p ~ v2 K Yj p p Rg Te (14) M j 1 j Ben-Gurion University of the Negev 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): YA YAL j A r R YA vs DA DL L (15) r r R r r R Other non-soluble components of the inert admixtures are not absorbed in the liquid J j 4R 2 j j 0, j 1, j A (16) Taking into account Eq. (16) and using anelastic approximation (Eq.12) we can obtain the expression for Stefan velocity: DL L YAL D1 Y1 vs (17) 1 Y1 r r R 1 Y1 r r R where subscript “1” denotes water vapor species Ben-Gurion University of the Negev Description of the model Stefan velocity and droplet vaporization rate The material balance at the gas-liquid interface yields: d mL dt 4 R 2 s vR, t R (18) Then assuming L we obtain the following expressions for the rate of change of droplet's radius (Elperin et al. 2005, 2007): DL YAL D1 Y1 R (19) 1 Y1 r r R L 1 Y1 r r R Ben-Gurion University of the Negev Description of the model Stefan velocity and droplet vaporization rate DL L YAL D1 Y1 vs 1 Y1 r r R 1 Y1 r r R DL YAL ρ D 1 Y1 R 1 Y1 r r R ρ L 1 Y1 r r R D1 Y1 vs In the case when all of the inert 1 Y1 r r R admixtures are not absorbed in liquid the expressions for Stefan D1 Y1 velocity and rate of change of R droplet radius read L 1 Y1 r r R Ben-Gurion University of the Negev Description of the model Stefan velocity and droplet vaporization rate Huckaby and Ray (1989) H 2O g DL 2 H 2O g L g R 2 t d R H 2O g 2 r t DLr YrA r g r 2 dt r L 1 Y1 r r R Tg g Tg L g R 2 t d R Tg 2 2 r t r r r g r 2 d t r for r Rt Ben-Gurion University of the Negev Description of the model Initial and boundary conditions The initial conditions for the system of equations (6)–(10) read: L At t = 0, 0 r R0 : T L T0 L Y AL YA,0 (20) At t = 0, r R0 : Y j Y j ,0 r Te Te,0 r At the droplet surface: Y j Dj Y j v s (21) r r R Y YAL YA v s D A A DL L (22) r r R r r R T dR T L YAL ke e L Lv kL La L DL (23) r r R dt r r R r r R Te R T L (24) R Ben-Gurion University of the Negev Description of the model Initial and boundary conditions The equilibrium between solvable gaseous and dissolved in liquid species reads: C S IV H * A p S ( IV ) A (25) where K1 K1K 2 (26) * K H 1 H S ( IV) 2 H H Huckaby & Ray (1989): 4 K 2 S IV 3 K1 12K 2 SO2 g K H S IV 2 2SO2 g K H 6 K 2 K1 2 K1K 2 S IV K H SO2 g K H SO2 g K H SO2 g 2 K1 4K 2 K1K 22 K H SO2 g 4K1K 2 K12 0 2 Ben-Gurion University of the Negev 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: 1, s p1, s Ts M 1 Y1, s Ts (27) pM The functional dependence of the Henry's law constant vs. temperature reads: K T H 1 1 ln H 0 (28) K H T RG T T0 Figure 2. Henry‟s constant vs. temperature Ben-Gurion University of the Negev Description of the model Initial and boundary conditions In the center of the droplet symmetry conditions yields: YAL T L 0 0 (27) r r r 0 r 0 At t 0 and r the „soft‟ boundary conditions at infinity are imposed: Y j Te 0 0 (28) r r r r Ben-Gurion University of the Negev Method of numerical solution Spatial coordinate transformation: r x 1 , for 0 r Rt ; Rt 1 r w 1, for r Rt ; Rt The gas-liquid interface is located at x w 0; w 0, 1 x 0, 1 Coordinates x and w can be treated identically in numerical calculations; Time variable transformation: DLt R0 ; 2 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: n i 1 xi i 1,, N 1 N Ben-Gurion University of the Negev Results and discussion YA L YA,L0 YA,Ls YA,L0 Average concentration of absorbed CO2 in the droplet: 1 YA L YAL r r 2 sin dr d dj Vd Analytical solution in the case of aqueous-phase controlled diffusion in a stagnant non-evaporating droplet: 6 1 1 n 2 exp 4 2n 2Fo 2 n1 Figure 3. Comparison of the numerical results with the experimental data (Taniguchi & Asano, 1992) DL t and analytical solution (Elperin et al 2005). Fo 4R 2 Ben-Gurion University of the Negev 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 Dependence concentration vs. time Figure 5.dioxide molarof dimensionless average aqueous SO2 concentration vs. time for various for various values of relative humidity (Elperin of evaporating initial sizes et al. 2005). droplet R0 (Elperin et al. 2005). Ben-Gurion University of the Negev Results and discussion Figure 8. Droplet surface temperature vs. time: Figure 7. Effect of Stefan flow and heat of 1 – model taking into account the equilibrium absorption on droplet surface temperature dissociation reactions; 2 – model of physical (Elperin et al. 2005). absorption (Elperin et al., 2007). Figure 6. Droplet surface temperature vs. time (Elperin et al., 2007). Ben-Gurion University of the Negev 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 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 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 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 Results and discussion: the interrelation between heat and mass transport Absorption during droplet evaporation Diffusion of Thermal effect of Reactions of absorbate absorption dissociation Decreases Stefan velocity Decreases vapor flux Increases droplet surface temperature Decreases effective Increases Henry’s constant vapor flux Decreases absorbate Decreases droplet flux surface temperature Increases effective Increases Stefan Henry’s constant velocity Increases absorbate flux Ben-Gurion University of the Negev 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

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