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T. Elperin, A. Fominykh and B. Krasovitov Department of Mechanical Engineering The Pearlstone Center for Aeronautical Engineering Studies

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Simultaneous Heat and Mass Transfer during Evaporation/Condensation on the Surface of a Stagnant Droplet in the Presence of Inert Admixtures Containing Non-condensable Solvable Gas: Application for the In-cloud Scavenging of Polluted Gases

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

Laboratory of Turbulent Multiphase Flowshttp://www.bgu.ac.il/me/laboratories/tmf/turbulentMultiphaseFlow.htmlHead - Professor Tov Elperin

People

Dr. Alexander Eidelman

Dr. Andrew Fominykh Mr. Ilia Golubev Dr. Nathan Kleeorin Dr. Boris Krasovitov Mr. Alexander Krein Mr. Andrew Markovich Dr. Igor Rogachevskii Mr. Itsik Sapir-Katiraie

- Motivation and goals
- Description of the model
- Gas absorption by stagnant evaporating/growing droplets
- Gas absorption by moving droplets
- Results and discussion: Application for the In-cloud Scavenging of Polluted Gases
- Conclusions

- SO2, CO2, CO – forest fires, volcanic emissions;
- NH3 – agriculture, wild animals
- ANTHROPOGENIC SOURCES
- SO2, CO2, CO – fossil fuels burning (crude oil and coal), chemical industry;
- NOx, CO2 – boilers, furnaces, internal combustion and diesel engines;
- HCl – burning of municipal solid waste (MSW) containing certain types of plastics

A diagram of the mechanism of polluted gases and aerosol flow through the atmosphere, their in-cloud precipitation and wet removal.

Gas absorption by stagnant droplets: Scientific background

- Dispersed-phase controlled isothermal absorption of a pure gas by stagnant liquid droplet (see e.g., Newman A. B., 1931);
- 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 solvable gas (see e.g., Ray A. K., Huckaby J. L. and Shah T., 1987, 1989)
- Simultaneous heat and mass transfer during evaporation/condensation on the surface of a stagnant droplet in the presence of inert admixtures containing non-condensable solvable gas (Elperin T., Fominykh A. and Krasovitov B., 2005)

Distance traveled by the polluted molecule

Diffusion of pollutant molecules through the gas

Vapor phase

Gas-liquid interface

Dissolution into the liquid at the interface

Liquid film

Diffusion of the dissolved species from the interface into the bulk of the liquid

Solution

= pollutant molecule

= pollutant captured in solution

Absorption equilibria

is the species in dissolved state

Henry’s Law

is the Henry’s Law constant

Aqueous phase sulfur dioxide/water chemical equilibria

Absorption of SO2 in water results in

(1)

The equilibrium constants for which are

The electroneutrality relation reads

(2)

Using the electroneutrality equation (11) and expressions for equilibrium constants (10) we obtain

(3)

where

is total dissolved sulfur in solution.

Z

Gas-liquid interface

q

R

Far field

Droplet

Y

j

X

Gas absorption by stagnant droplet

Description of the model

Governing equations

1. gaseous phase r > R (t)

(4)

(5)

(6)

2. liquid phase 0 < r < R (t)

(7)

(8)

In Eqs. (5)

- subsonic flow velocities (low Mach number approximation, M << 1)

(9)

(10)

In spherical coordinates Eq. (9) reads:

The radial flow velocity can be obtained by integrating equation (10):

(11)

(12)

Stefan velocity and droplet vaporization rate

The continuity condition for the radial flux of the absorbate at the droplet surface reads:

(13)

Other non-solvable components of the inert admixtures are not absorbed in the liquid

(14)

Taking into account this condition and using Eq. (10) we can obtain the expression for Stefan velocity:

(15)

where subscript “1” denotes water vapor species

Stefan velocity and droplet vaporization rate

The material balance at the gas-liquid interface yields:

(16)

Then assuming we obtain the following expression for the rate of change of droplet\'s radius:

(17)

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

Initial and boundary conditions

The initial conditions for the system of equations (1)–(5) read:

(18)

At t = 0,

At the droplet surface the continuity conditions for the radial flux of non-solvable gaseous species yield:

(19)

For the absorbate boundary condition reads:

(20)

The droplet temperature can be found from the following equation:

(21)

At and the ‘soft’ boundary conditions at infinity are imposed

Initial and boundary conditions

The equilibrium between solvable gaseous and dissolved in liquid species can be expressed using the Henry\'s law

(22)

At the gas-liquid interface

(23)

In the center of the droplet symmetry conditions yields:

(24)

(25)

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:

(26)

where

The functional dependence of the Henry\'s law constant vs. temperature reads:

(27)

Fig. 1. Henry\'s law constant for aqueous solutions of different solvable gases vs. temperature.

- 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 (4)–(8) was solved using the method of lines;
- The mesh points are spaced adaptively using the following formula:

Fig. 2. Temporal evolution of radius of evaporating water droplet in dry still air. Solid line – present model, dashed line – non-conjugate model (Elperin & Krasovitov, 2003), circles – experimental data (Ranz & Marshall, 1952).

Fig. 3. Comparison of the numerical results with the experimental data (Taniguchi & Asano, 1992) and analytical solution.

Average concentration of absorbed CO2 in the droplet:

Analytical solution in the case of aqueous-phase controlled diffusion in a stagnant non-evaporating droplet:

Fig. 5. Dependence of average aqueous SO2

molar concentration vs. time

Fig. 4. Dependence of average aqueous CO2

molar concentration vs. time

Typical atmospheric parameters

Table 1. Observed typical values for the radii of cloud droplets

Fig. 6. Vertical distribution of SO2. Solid lines - results of calculations with (1) an 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)

Fig. 7. Dependence of dimensionless

average aqueous CO2 concentration vs.

time (RH = 0%).

Fig. 8. Dependence of dimensionless

average aqueous SO2 concentration vs.

time (RH = 0%).

Fig. 9. Dependence of dimensionless average

aqueous CO2 concentration vs. time

(R0 = 25 mm).

Fig. 11. Effect of Stefan flow and heat of

absorption on droplet surface temperature.

Fig. 10. Droplet surface temperature vs. time

(T0 = 274 K, T∞ = 288 K).

Fig. 12. Droplet surface temperature N2/CO2/H2O

gaseous mixture (YH2O = 0.011).

Fig. 13. Droplet surface temperature N2/SO2

gaseous mixture.

Fig. 14. Droplet surface temperature N2/NH3

gaseous mixture.

Fig. 15. Dimensionless droplet radius vs. time

R0 = 25 mm, XSO2 = 0.1 ppm.

Fig. 16. Dimensionless droplet radius vs. time

R0 = 100 mm, N2/CO2 gaseous mixture.

Fig. 17. Dimensionless droplet radius vs. time

N2/CO2/H2O gaseous mixture YH2O = 0.011.

Fig. 18. Dimensionless droplet radius vs. time

N2/CO2/H2O gaseous mixture.

Conjugate Mass Transfer during Gas Absorption

by Falling Liquid Droplet with Internal Circulation

Developed model of solvable gas absorption from the mixture with inert gas by falling droplet (Elperin & Fominykh, Atm. Evironment 2005) yields the following Volterra integral equation of the second kind for the dimensionless mass fraction of an absorbate in the bulk of a droplet:

(28)

where - dimensionless mass

fraction of an absorbate in the bulk of a droplet;

- droplet Peclet number;

- initial value of mass fraction of absorbate in a droplet;

- mass fraction in the bulk of a gas phase;

- dimensionless thickness of a diffusion boundary layer inside a droplet;

k - relation between a maximal value of fluid velocity at droplet interface

to velocity of droplet fall;

- dimensionless time.

Fig. 19. Dependence of the concentration of the

dissolved gas in the bulk of a water droplet 1-Xb

Vs. time for absorption of CO2 by water in the

presence of inert admixture.

Fig. 20. Dependence of the concentration of

the dissolved gas in the bulk of a water droplet

1-Xb vs. time for absorption of SO2 by water in

the presence of inert admixture.

Heat and mass transfer on the surface of moving droplet at small Re and Pe numbers

Heat and mass fluxes extracted/delivered from/to the droplet surface (B. Krasovitov and E. R. Shchukin, 1991):

(29)

(30)

Where

- dimensionless concentration;

- Peclet number.

In this study we developed a model that takes into account the simultaneous effect of gas absorption and evaporation (condensation) for a system consisting of liquid droplet - vapor of liquid droplet - inert noncondensable and nonabsorbable gas-noncondensable solvable gas.

Droplet evaporation rate, droplet temperature, interfacial absorbate concentration and the rate of mass transfer during gas absorption are highly interdependent.

Thermal effect of gas dissolution in a droplet and Stefan flow increases droplet temperature and mass flux of a volatile species from the droplet temperature at the initial stage of evaporation.

The obtained results show good agreement with the experimental data .

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.

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