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

http://www.bgu.ac.il/me/laboratories/tmf/turbulentMultiphaseFlow.html
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

A diagram of the mechanism of polluted gases and aerosol flow through the atmosphere, their in-cloud precipitation and wet removal.
NATURAL SOURCES
 SO
2
, CO2, CO – forest fires, volcanic emissions;
 NH
3
– 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

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)

Vapor phase
Gas-liquid interface
Liquid film
Solution
Absorption equilibria
Diffusion of pollutant molecules through the gas
Dissolution into the liquid at the interface
A
  
H
2
O A

H
2
O
A

H
2
O is the species in dissolved state
Diffusion of the dissolved species from the interface into the bulk of the liquid
Henry’s Law

A

H
2
O


H
A p
A
H
A is the Henry’s Law constant
= pollutant molecule
= pollutant captured in solution

Aqueous phase sulfur dioxide/water chemical equilibria
Absorption of SO
2 in water results in
SO
2

H
2
O
SO
2

H
2
O
SO
2

H
2
O
H
 
HSO

3
HSO

3
H
2
O
H
 
SO
3
2

H
 
OH

(1)
K
H
The equilibrium constants for which are


SO
2 p

H
SO
2
2
O

K
1


  
SO
2

HSO
3

H
2
O


K
2

   

HSO
3

 3
2

K w

The electroneutrality relation reads
    
HSO

3
  
3

(2)

Huckaby & Ray (1989)
  
HSO

3
  
3
2

Using the electroneutrality equation (11) and expressions for equilibrium constants (10) we obtain
4 K
2

S

3 

 K
1

12 K
2


SO
2
K
 
2 w

K
H
K
1



SO
2

K
H

S
  
2 
2




SO
2

S

K
H


K
H

6 K
2
SO
2



K
1
 
K
H

2 K
1
K
2
SO
2


K

2
H
K
H
K

2
K
SO
2 w

SO
2


2

K
1

K
H
K
1

K
2 w
SO
2

4 K
2
 
K
1
K
2
2






K
H

SO
2


4 K
1
K
2

K
1
2


K
 w
K
1

2 K
2
 
K w
2
K
1




0 at r

R
 
(3) where

S
 
SO
2

H
2
O



HSO

3
  
3
2
 is total dissolved sulfur in solution.

Governing equations r r
2
2
1. gaseous phase r > R ( t )


 t
  j

 c p
T e

 t r

2




 t r


 r




 v
 r
 v r r r r

2 r
Y
2 c
2 j


 p
T e v r




 r





0
 r


D k e j r r
2
2

Y j
 r

T e
 r




2. liquid phase 0 < r < R ( t ) r
2

T
 t
 
L 
 r

 r
2

T
 r

 r
2

 t


L
Y
A



 r



L
D
L r
2

Y

(
A r
L )


(4)
(5)
(6)
(7)
(8)
X
Droplet

A
Z j q ds
Far field d

Gaseous phase

L
R
In Eqs. (5) j

1 ,..., K

1 ,
 j
K

1
Y j

1
Gasliquid interface
Y

anelastic approximation: v 2 c
2 
1 Eq .
4
    
0 .
In spherical coordinates Eq. (9) reads:

 r r

2  v r


0
(9)
(10)
The radial flow velocity can be obtained by integrating equation (10):
 v r r
2  const (11) subsonic flow velocities (low Mach number approximation, M << 1)
 p ~
 v 2 p
 p

 
R g
T e j
K


1
Y j
M j
(12)

Stefan velocity and droplet vaporization rate
The continuity condition for the radial flux of the absorbate at the droplet surface reads: j
A r

R
 
Y
A v s

D
A


Y
A
 r r

R 
 
D
L

L

Y

A r r

R 
(13)
Other non-solvable components of the inert admixtures are not absorbed in the liquid
J j

4

R
2 j j

0 , j

1 , j

A (14)
Taking into account this condition and using Eq. (10) we can obtain the expression for Stefan velocity: v s
 
 
D
L
1


L
Y
1


Y

A r r

R 


1
D
1

Y
1


Y
1
 r r

R 
(15) where subscript
“
1
” denotes water vapor species

Stefan velocity and droplet vaporization rate
The material balance at the gas-liquid interface yields: d m
L d t
 
4

R
2  s
 v
R ,



L
  rate of change of droplet's radius:


1
D
L

Y
1


Y

A r r

R 


L


1
D

1
Y
1


Y
1
 r r

R 
(16)
(17)

Stefan velocity and droplet vaporization rate v s
 
 
D
1
L


L
Y
1


Y

A r


1
D
L

Y
1


Y

A r r

R 


1
D
1

Y
1


Y
 r
1 r

R  r

R 

ρ
L

ρ
1
D

1
Y
1


Y
 r
1 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 v s
 

1
D
1

Y
1


Y
1
 r r

R 


L


1
D

1
Y
1


Y
1
 r r

R 

Initial and boundary conditions
The initial conditions for the system of equations (1)–(5) read:
At t = 0, 0
 r

R
0
: T

T
0
Y
 
A

Y
A , 0
At t = 0, r

R
0
: Y j

Y j , 0
T e

T e , 0
(18)
At the droplet surface the continuity conditions for the radial flux of nonsolvable gaseous species yield:
D j


Y j
 r r

R 
 
Y j v s
(19)
For the absorbate boundary condition reads:

Y
A v s

D
A


Y
 r
A r

R 
 
D
L

L

Y

A r r

R 
(20)
The droplet temperature can be found from the following equation: k e

T e
 r r

R 
 
L
L v d R d t
 k
L

T
 r r

R 

L a

L
D
L

Y

A r r

R 
(21)

Initial and boundary conditions
The equilibrium between solvable gaseous and dissolved in liquid species can be expressed using the Henry's law
C
A

H
A p
A
(22)
At the gas-liquid interface
T e

T
In the center of the droplet symmetry conditions yields:
(23) t

0 r

Y

A r r

0

0

T
 r r

0

0
(24)
 ‘ soft ’ boundary conditions at infinity are imposed

Y j
 r r
 

0

T e
 r r
 

0 (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 T s
( t ) and can be determined as follows: where p
 p
Y
1 ,
 s
  
Y
1 , s
  

1 ,
 s  p
1 , s
  p M
M
1
(26)
The functional dependence of the Henry's law constant vs. temperature reads: ln
H
H
A
A
 
0
 


H
R
G


1
T

1
T
0

 (27)
Fig. 1. Henry's law constant for aqueous solutions of different solvable gases vs. temperature.

Spatial coordinate transformation: x w


1
1





R
R r
  r
 
,

1


, for for
0 r

 r
R

 
;
R
 
;
The gas-liquid interface is located at x
 w

0 ; w
 x
 numerical calculations;

Coordinates x and w can be treated identically in
Time variable transformation:
 
D
L t R
0
2
;
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: x i

N
1 n i

1 ,  , N

1

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).

 
Y
A
Y
A , s

Y
A , 0

Y
A , 0
Y
Average concentration of absorbed
CO
2 in the droplet:
A

1
V d

Y
A r
2 sin
 dr d
 d j
Analytical solution in the case of aqueous-phase controlled diffusion in a stagnant non-evaporating droplet:
 
1


6
2

 n

1
1 n
2 exp


4
 2 n
2
Fo

Fo

D
L t
D d
Fig. 3.
Comparison of the numerical results with the experimental data (Taniguchi &
Asano, 1992) and analytical solution.

Fig. 4.
Dependence of average aqueous CO
2 molar concentration vs. time
Fig. 5.
Dependence of average aqueous SO
2 molar concentration vs. time

Typical atmospheric parameters
Table 1 . Observed typical values for the radii of cloud droplets
Cloudtype/particle type stratus cumulus cumulonimbus growing cumulus fog orographic drizzle
Rain drops
Droplet Radius
4.7 – 6.7 m
3 – 5 m m
6 – 8 m m m
Reference
E. Linacre and B.
Geerts (1999)
~20 m m
8 m m – 0.5 mm up to 80 m
~ 1.2 mm m
0.1 – 2.0 mm
Cooperative
Convective
Precipitation
Experiment (CCOPE)
University of
Wyoming
E. Linacre and B.
Geerts (1999)
H. R. Pruppacher and
J. D. Klett (1997)
–
–
Fig. 6.
Vertical distribution of SO
2
.
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 CO
2 concentration vs.
time (RH = 0%).
Fig. 8.
Dependence of dimensionless average aqueous SO
2 concentration vs.
time (RH = 0%).
Fig. 9.
Dependence of dimensionless average aqueous CO
2
(R
0
= 25 m m).
concentration vs. time

Fig. 10.
Droplet surface temperature vs. time
(T
0
= 274 K, T
∞
= 288 K).
Fig. 11.
Effect of Stefan flow and heat of absorption on droplet surface temperature.

Fig. 12.
Droplet surface temperature N
2
/CO
2
/H
2
O gaseous mixture (Y
H2O
= 0.011).
Fig. 13.
Droplet surface temperature N
2
/SO
2 gaseous mixture.
Fig. 14.
Droplet surface temperature N
2
/NH
3 gaseous mixture.

Fig. 15.
Dimensionless droplet radius vs. time
R
0
= 25 m m, X
SO2
= 0.1 ppm.
Fig. 16.
Dimensionless droplet radius vs. time
R
0
= 100 m m, N
2
/CO
2 gaseous mixture.
Fig. 17.
Dimensionless droplet radius vs. time
N
2
/CO
2
/H
2
O gaseous mixture Y
H2O
= 0.011.
Fig. 18.
Dimensionless droplet radius vs. time
N
2
/CO
2
/H
2
O 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
X b
(

)

1

Pe
L

( 1
3

H
A

D )


X b
(

0 b
(

)
  x b
( t )

H
A x
2
(

)

 x
 
L
0 
H
A x
2
)


0

(

)
L

( q sin
,
 q
 
) d

(28) fraction of an absorbate in the bulk of a droplet;
Pe

L
UkR

L x
2 x
L
(

)
 
L
/
D
L
R k
- 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;
- relation between a maximal value of fluid velocity at droplet interface to velocity of droplet fall;
  tUk R - dimensionless time.

Fig. 20. Dependence of the concentration of the dissolved gas in the bulk of a water droplet
1-X b vs. time for absorption of SO
2 the presence of inert admixture.
by water in
Fig. 19.
Dependence of the concentration of the dissolved gas in the bulk of a water droplet 1-X b
Vs. time for absorption of CO presence of inert admixture.
2 by water in the

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):
Where
Pe
 c
1
 n
1
Pe
T n

Pe
D
J
T
J m


4

J
T
R 1
 c
1 , s
Pe
4
  i

T s

T  k e dT e c
1 ,


T s

T  k e n D
1 dT e
- dimensionless concentration;
- Peclet number.
(29)
(30)
Pe
T

U



R
Pe
D

U

R
D
1

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.