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Report
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 Flows
http://www.bgu.ac.il/me/laboratories/tmf/turbulentMultiphaseFlow.html
Head - 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
Outline of the presentation
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
NATURAL SOURCES
 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
Absorption equilibria
Vapor phase
Gas-liquid interface
Diffusion of pollutant
molecules through
the gas
Dissolution into the
liquid at the interface
= pollutant molecule
= pollutant captured in solution
A  H 2O
A  H 2O is the species in
dissolved state
Henry’s Law
Liquid film
Solution
A g   H 2O
Diffusion of the
dissolved species
from the interface
into the bulk of the
liquid
 A  H 2O   H A p A
H A is the Henry’s Law
constant
Aqueous phase sulfur dioxide/water chemical equilibria
Absorption of SO2 in water results in
SO2  g   H 2O
SO2  H 2O
HSO3
H 2O
SO2  H 2O
H   HSO3
H

H   OH 
The equilibrium constants for which are
KH

SO2  H 2O 

pSO2
H  HSO 


K1
(1)
 SO32

3
SO2  H 2O

H  SO 

HSO 

K2
2
3

3
 
K w  H  OH 
The electroneutrality relation reads
H   OH  HSO  2SO 



3
2
3
(2)

Huckaby & Ray (1989)
H   HSO  2SO 


3
2
3
Using the electroneutrality equation (11) and expressions for equilibrium
constants (10) we obtain


K w2
 SO2  g  K H S IV  2 
4 K 2 S IV    K1  12K 2 





SO
g
K
K
2
H
1




K2Kw
K w2
2SO2  g K H 6 K 2  K1   2 K1K 2 


K H SO2  g  K H K1SO2  g 

3
S IV K H SO2  g   K H SO2  g K H2 SO2  g 2 K1  4 K 2   K1K 22 

K H SO2  g 
where
4 K1K 2  K12

K w2 
 K w  K1  2 K 2  
0
K1 
S IV   SO2  H 2O  HSO3 
is total dissolved sulfur in solution.

 SO32
at r  Rt 

(3)
Gas absorption by stagnant droplet
Description of the model
Governing equations
1. gaseous phase r > R (t)
  2
r2

r  vr  0
t r
Y 


 
r 2 Y j  
 vr r 2Y j    D j r 2 j 
t
r
r 
r 
  c pTe  
T 

r2

 vr r 2c pTe   ke r 2 e 
t
r
r 
r 
2. liquid phase 0 < r < R (t)
 L
  2 T  L  
2 T

r
  L  r
t
r 
r 
( L) 


Y


r 2  LYA L     L DL r 2 A 
t
r 
r 








Droplet
Far field
(4)
Gaseous phase
Z
ds
(5)
 m A
(6)
 m L
d
q
R
Gasliquid
interface
Y
(7)
(8)
X
j
In Eqs. (5)

K
Y
j 1 j
j  1,..., K  1,
1
anelastic approximation:
v 2 c 2  1 Eq. 4 
 v   0.
(9)


 2
(10)
r  vr  0
r
The radial flow velocity can be obtained by integrating equation (10):
In spherical coordinates Eq. (9) reads:
 vr r 2  const
(11)
subsonic flow velocities (low Mach number approximation, M << 1)
 p ~  v2
 Yj 

p  p  Rg Te  
M 
j
j 1 
K
(12)
Stefan velocity and droplet vaporization rate
The continuity condition for the radial flux of the absorbate at the droplet
surface reads:
YA L 
YA
j A r  R  YA vs  DA 
  DL  L
(13)
r r  R
r r  R


Other non-solvable components of the inert admixtures are not absorbed in the
liquid
(14)
J j  4R 2 j j  0,
j  1, j  A
Taking into account this condition and using Eq. (10) we can obtain the
expression for Stefan velocity:
DL  L YA L 
vs  
 1  Y1  r

r  R
D1 Y1
1  Y1  r r  R
where subscript “1” denotes water vapor species

(15)
Stefan velocity and droplet vaporization rate
The material balance at the gas-liquid interface yields:

d mL
 4 R 2  s vR, t   R
dt

(16)
Then assuming  L   we obtain the following expression for the
rate of change of droplet's radius:
 L

Y
D
L
A
R 
1  Y1  r

r  R
 D1
Y1
 L 1  Y1  r r  R

(17)
Stefan velocity and droplet vaporization rate
DL  L YA L 
vs  
 1  Y1  r
 L

Y
D
L
A
R 
1  Y1  r

r  R

r R
D1 Y1
1  Y1  r r  R

ρ D 1 Y1
ρ L 1  Y1  r r  R
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

D1 Y1
vs  
1  Y1  r r  R

R 
 D1
Y1
 L 1  Y1  r r  R

Initial and boundary conditions
The initial conditions for the system of equations (1)–(5) read:
At t = 0, 0  r  R0 :
At t = 0,
r  R0 :
T  L   T0 L 
 L
 
Y AL  YA,0
Y j  Y j ,0 r 
Te  Te,0 r 
(18)
At the droplet surface the continuity conditions for the radial flux of nonsolvable gaseous species yield:
Y j
(19)
Dj
 Y j v s
r r  R

For the absorbate boundary condition reads:
YA L 
YA
YA v s  D A 
  DL  L
r r  R
r

(20)
r  R
The droplet temperature can be found from the following equation:
T
ke e
r
r  R
dR
T  L 
  L Lv
 kL
dt
r
r  R
YA L 
 La  L DL
r
(21)
r  R
Initial and boundary conditions
The equilibrium between solvable gaseous and dissolved in liquid species
can be expressed using the Henry's law
CA  H A pA
(22)
Te  T  L 
(23)
At the gas-liquid interface
In the center of the droplet symmetry conditions yields:
YA L 
r
0
r 0
T  L 
r
0
(24)
r 0
At t  0 and r   the ‘soft’ boundary conditions at infinity are imposed
Y j
r
0
r 
Te
r
0
r 
(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:
1, s p1, s Ts  M 1
Y1, s R, t   Y1, s Ts  


pM
where p  p
(26)
The functional dependence of the Henry's law constant vs. temperature reads:
ln
H A T0   H  1 1 
  

H A T  RG  T T0 
Fig. 1. Henry's law constant for aqueous
solutions of different solvable gases vs.
temperature.
(27)
Method of numerical solution
Spatial coordinate transformation:
r
x  1
,
for 0  r  Rt ;
Rt 
1 r

w 
 1, for r  Rt ;
  Rt  
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;
  DLt R02 ;
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:
 i  1
xi  

 N 
n
i  1,, N  1
Results and discussion
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).

YA L   YA ,L0
YA ,Ls  YA ,L0
Average concentration of absorbed
CO2 in the droplet:
YA L 
1
  YA L  r  r 2 sin  dr d dj
Vd
Analytical solution in the case of
aqueous-phase controlled diffusion
in a stagnant non-evaporating
droplet:
  1
6


1
2 2
exp

4

n Fo
2 2
 n1 n
Fo 
DLt
Dd

Fig. 3. Comparison of the numerical results
with the experimental data (Taniguchi &
Asano, 1992) and analytical solution.
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
Cloudtype/particle type
stratus
Droplet Radius
4.7 – 6.7 mm
cumulus
3 – 5 mm
cumulonimbus
6 – 8 mm
growing cumulus
fog
orographic
drizzle
Rain drops
Reference
~20 mm
8mm – 0.5 mm
up to 80 mm
E. Linacre and B.
Geerts (1999)
Cooperative
Convective
Precipitation
Experiment (CCOPE)
University of
Wyoming
E. Linacre and B.
Geerts (1999)
H. R. Pruppacher and
J. D. Klett (1997)
~ 1.2 mm
–
0.1 – 2.0 mm
–
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 (YH O = 0.011).
2
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. 17. Dimensionless droplet radius vs. time
N2/CO2/H2O gaseous mixture YH2O = 0.011.
Fig. 16. Dimensionless droplet radius vs. time
R0 = 100 mm, N2/CO2 gaseous mixture.
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:
X b ( )  1 

PeL

3
sin q
X
(

)
d
b


 (q ,   )
 (1  H A  D) 0
0 L
where X b ( )   xb (t )  H A x2 () 
x   H A x2 ()
0
L
(28)
- dimensionless mass
fraction of an absorbate in the bulk of a droplet;
Pe L  UkR DL
x0
L
x 2 ( )
- droplet Peclet number;
- initial value of mass fraction of absorbate in a droplet;
- mass fraction in the bulk of a gas phase;
L  L / R
- 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;
  tUk R
- 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):
Ts
 Pe 
J T  4 R1    k e dTe
4 T


J m  JT
c1,s Ti   c1, 
Ts

T
Where
n1
n
Pe  PeT  Pe D
c1 
PeT 
(30)
ke
dTe
nD1
- dimensionless concentration;
- Peclet number.
UR

Pe D 
(29)
UR
D1
Conclusions
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 gasnoncondensable 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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