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
Motivation and goals
Fundamentals
Description of the model
Results and discussion
Conclusions
Summer Heat Transfer Conference
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Ben-Gurion University of the Negev
Atmospheric polluted
gases (SO2, CO2, CO,
NOx, NH3):
Scavenging of air pollutions
by cloud and rain droplets
• In-cloud scavenging
of polluted gases
• Scavenging of air
pollutions by rain
droplets
Single Droplet
Air
Soluble gas
Summer Heat Transfer Conference
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Henry’s Law:
is the species in
dissolved state
Ben-Gurion University of the Negev
Gaseous pollutants in atmosphere
Scavenging of
air pollutions
SO2 and NH3 – anthropogenic
emission
CO2 – competition between
photosynthesis, respiration and
thermally driven buoyant
mixing
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Fig. 1a. Aircraft observation of vertical profiles
of CO2 concentration (by Perez-Landa et al.,
2007)
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Gaseous pollutants in atmosphere
Scavenging of
air pollutions
SO2 and NH3 – anthropogenic
emission
CO2 – competition between
photosynthesis, respiration and
thermally driven buoyant
mixing
Summer Heat Transfer Conference
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Fig. 1b. Vertical distribution of SO2.
Solid lines - results of calculations with
(1) and 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)
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Vertical temperature profile in the
lowest few kilometers of the
atmosphere
Scavenging of
air pollutions
Adiabatic decrease of atmospheric
temperature with height
Inversion of vertical temperature
gradient as a result of solar
radiation heating and ground
cooling
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Fig. 1c. Aircraft observation of
potential temperature vs. height (by
Perez-Landa et al., 2007)
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Gas absorption by falling droplets:
• Walcek and Pruppacher, 1984
• Alexandrova et al., 2004
• Elperin and Fominykh, 2005
Measurements of vertical distribution of trace gases in the
atmosphere:
• SO2 – Gravenhorst et al., 1978
• NH3 – Georgii & Müller, 1974
• CO2 – Denning et al., 1995; Perez-Landa et al., 2007
Scavenging of gaseous pollutants by falling rain droplets in
inhomogeneous atmosphere:
• Elperin, Fominykh & Krasovitov 2008 – non-uniform
concentration distribution in a gaseous phase
• Elperin, Fominykh & Krasovitov 2009 – non-uniform temperature
and concentration distribution in the atmosphere
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In the analysis we used the following
assumptions:
dc << R; dT << R
Tangential molecular mass transfer rate
along the surface is small compared with a
molecular mass transfer rate in the normal
direction
The bulk of a droplet, beyond the diffusion
and temperature boundary layers, is
completely mixed by circulations inside a
0.1 mm  R  0.5 mm
droplet
 300 shape.
The droplet 10
has aRe
spherical
0.7  U  4.5 m/s
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Fig. 2. Schematic view of a falling
droplet and temperature and
concentration profiles
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System of convective diffusion and energy conservation transient equations for the
liquid and gaseous phases read:
 sin  xi 2 y cos  xi 
 xi
 2 xi
 U  k 



,
  Di
2
t
R 
R
y 
y

 sin  Ti 2 y cos Ti 
Ti
 2Ti
 U  k 

  ai 2
t
R
y 
y
 R 
(1)
(i = 1, 2)
Fluid velocity components at the gas-liquid interface are (Prippacher & Klett, 1997):
v  kU sin 
2kU
vr 
y cos
R
(2)
where k = 0.009  0.044 for different Re, and y  R.
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Initial and boundary conditions
Using the transformation z = U t the coordinate-dependent boundary conditions
can be transformed into the time-dependent boundary conditions:
x2  xb 2 (t ) at
y
(3)
T2  Tb2 (t ) at
y
x1  xb1 (t ) at
y   (4)
T1  Tb1(t ) at
y   (8)
x1  m x2
at
y0
(5)
T1  T2
at
y0
(9)
N D1  N D 2 at
y0
(6)
NT 1  NT 2 at
y0
(10)
where N Di   Di Ci
(7)
T
 xi
, NTi  i i , m  H A RgTC2 C1 , y  r  R
y
y
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Introduction of the self-similar variables: Ti 
y
,
dTi t , 
 Di 
y
d Di t , 
and application of Duhamel's theorem yields a solution of convection diffusion
and energy conservation equations (Eqs. 1):

i (Y , , ) 

i 1






a
 b,2 ( )  b,1 ( )
Y
i 1 T

 d
b,i ( )   1
erfc
  ( ,   ) 
1 T a

 T,i

0


(11)

X i (Y , , ) 





Y
i 1  D D  X b1 ( )  m s X b 2 ( )

 d
erfc
 X b,i ( )   1
1  m s  D D
 D,i ( ,   ) 


0


1  f ( , ) 1  1  f ( , ) 3  
1
3
where
2i 
cos(

)

cos
( )  
 
  ,

4
3
1

f
(

,

)
3
1

f
(

,

)
Pei sin ( ) 


  
 
f ( , )  tg 2   exp(2 ) and   tUk / R, i  Ti / T10 , X1(t )  x1(t ) / m0 T20 x20 ,
2
X 2 (t )  x2 (t ) / x20
4
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Integral energy and material balances over the droplet yields:

db1
3
1

sin  d ,
d
2  PeT 1 Y Y 0

0
d X b1
3
X 1

sin  d
d
2  PeD1 Y Y 0
Substituting solutions (11) into Eqs. (12) yields:
b1 (t )  1 
3
  PeT1 (1   T a )




0

 b2 ( )  b1( ) 
0
0
sin  d d
T1 ( ,   )
where xb10
xb 20
(13)

 X b1 ( )  m( s ( )) X b 2 ( )  sin  d d
x
3
X b1 ( )  b10 
  ( ,   )
m0 xb 20
  PeD 1 
1  m( s ( ))   D
 0 D1
0

(12)

(14)
 initial value of molar fraction of absorbate in a droplet
 value of molar fraction of an absorbate in a gas phase at height H
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The system of equations for temperature and absorbate concentration in the bulk
of a droplet:



 X b1 ( )  m( s ( )) X b 2 ( )  sin  d d
xb10
3
 X b1 ( ) 

  ( ,   )
m0 xb 20
  PeD 1 
1  m( s ( ))   D

 0 D1
0



3
sin  d d




(
t
)

1


(

)


(

)
b1
b2
b1

  PeT1 (1   T a )
 ( ,   )

0
0 T1




is a system of linear convolution Volterra integral equations of the second
kind that can be written in the following form:


 
f    f   K  ,   d  g ,   0, t 
(15)
0
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The method of solution is based on the approximate calculation of a definite
integral using some quadrature formula:
b
N
 F   d  
a
i
F i   RN [ F ], i  a,b, i  1, 2, ..., N ,
(16)
i 1
where RN F  – remainder of the series after the N-th term.
T  T0
The uniform mesh with an increment h was used: Ti  T0  ih, h  N
N
Using trapezoidal integration rule we obtain a system of linear algebraic
equations:
f (0)  g (0),
i 1
1
1
(1  h  K ii )  fi  h ( K i 0 f 0   K i j  f j )  gi , i  1,, N
2
2
j 1
(17)
The kernel K  ,   is a matrix M  M and equation (17) is viewed as a vector
equation.
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Fig. 3. Dependence of CO2 concentration in
the atmosphere vs. altitude (1) aircraft
measurements Valencia 6:23 (by Perez-Landa
et al., 2007); (2)-(4) approximation of the
measured data; (3) aircraft measurements
Valencia 13:03 (by Perez-Landa et al., 2007).
Fig. 4. Dependence of the potential,
atmospheric and droplet surface
temperature vs. altitude in the morning.
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Fig. 3. Dependence of CO2 concentration in
the atmosphere vs. altitude (1) aircraft
measurements Valencia 6:23 (by Perez-Landa
et al., 2007); (2)-(4) approximation of the
measured data; (3) aircraft measurements
Valencia 13:03 (by Perez-Landa et al., 2007).
Fig. 5. Dependence of the potential,
atmospheric and droplet surface
temperature vs. altitude in the afternoon
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Fig. 3. Dependence of CO2 concentration in
the atmosphere vs. altitude (1) aircraft
measurements Valencia 6:23 (by Perez-Landa
et al., 2007); (2)-(4) approximation of the
measured data; (3) aircraft measurements
Valencia 13:03 (by Perez-Landa et al., 2007).
Fig. 6. Dependence of the concentration of the
dissolved CO2 gas in the bulk of a falling rain
droplet vs. time, xb10 = 0.
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Fig. 8. Dependence of the relative
concentration of ammonia (NH3)
inside a water droplet vs. time.
Fig. 7. Dependence of the interfacial
temperature of a falling rain droplet vs.
altitude.
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Fig. 9. Evolution of ammonia
(NH3) distribution in the
atmosphere due to scavenging
by rain
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The suggested model of gas absorption by a falling liquid droplet in the
presence of inert admixtures takes into account a number of effects that
were neglected in the previous studies, such as the effect of dissolved
gas accumulation inside a droplet and effect of the absorbate and
temperature inhomogenity in a gaseous phase on the rate of heat and
mass transfer.
It is shown than if concentration of a trace gas in the atmosphere is
homogeneous and temperature in the atmosphere decreases with height,
beginning from some altitude gas absorption is replaced by gas
desorption.
We found that the neglecting temperature inhomogenity in the
atmosphere described by adiabatic lapse rate leads to overestimation of
trace gas concentration in a droplet at the ground on tens of percents.
If concentration of soluble trace gas is homogeneous and temperature
increases with height e.g. during the nocturnal inversion, droplet
absorbs gas during all the time of its fall.
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