Paper No 7th International Conference on Multiphase Flow
ICMF 2010, Tampa, FL USA, May 30June 4, 2010
Effect of concentration and temperature inhomogeneity in a gaseous phase on gas
absorption by falling liquid droplets
Tov Elperin, Boris Krasovitov and Andrew Fominykh
Department of Mechanical Engineering
The Pearlstone Center for Aeronautical Engineering Studies
BenGurion University of the Negev P. O. B. 653, 84105, Israel
Email: elperin@bgu.ac.il, borisk@bgu.ac.il, f:miim Li i;,u .c i
Keywords: precipitation scavenging, trace gases, rain droplets, gas absorption, heat transfer, mass transfer
Abstract
We analyze nonisothermal absorption of trace gases by the rain droplets with internal circulation which is caused by interfacial
shear stresses. It is assumed that the concentration of soluble trace gases and temperature in the atmosphere varies in a vertical
direction. The rate of scavenging of soluble trace gases by falling rain droplets is determined by solving heat and mass transfer
equations. In the analysis we accounted for the accumulation of the absorbate in the bulk of the falling rain droplet. The problem
is solved in the approximation of a thin concentration and temperature boundary layers in the droplet and in the surrounding air.
We assumed that the bulk of a droplet, beyond the diffusion boundary layer, is completely mixed and concentration of the
absorbate and temperature are homogeneous and timedependent in the bulk. By combining the generalized similarity
transformation method with Duhamel's theorem, the system of transient conjugate equations of convective diffusion and energy
conservation for absorbate transport in liquid and gaseous phases with timedependent boundary conditions is reduced to a
system of linear convolution Volterra integral equations of the second kind which is solved numerically. Calculations are
performed using available experimental data on concentration and temperature profiles in the atmosphere. 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. Neglecting temperature inhomogenity in the
atmosphere described by adiabatic lapse rate leads to essential overestimation of the trace gas concentration in a droplet on the
ground.
Introduction
Gas absorption by the falling rain droplets is of
relevance in meteorology and environmental engineering.
Rains play an important role in wet removal of gaseous
pollutants from the atmosphere. Furthermore, scavenging of
soluble hazardous gases by rains is an important part of
selfcleansing process of the atmosphere (see, e.g.
Flossmann, 1998 and Pruppacher and Jaenicke, 1995).
Scavenging of atmospheric gaseous pollutants by rain
droplets is a result of a gas absorption mechanism
(Pruppacher and Klett, 1997). Comprehensive study of mass
transfer during gas absorption by falling rain droplets is also
required for predicting transport of hazardous gases in the
atmosphere.
Concentration measurements of C02, SO2 NH3 and
other trace gases in the atmospheric boundary layer revealed
vertical (altitudinal) dependence of the concentrations (see
Georgii, 1978; Gravenhorst et al., 1978; Georgii and Miiller,
1974; Denning et al., 1995). Concentration of gases which
are not associated with photosynthesis, e.g. SO2 and
NH3, has a maximum at the Earth surface and decreases
with height over the continents. The concentration of NH3
over the continents decreases rapidly with altitude, reaching
a constant background concentration in winter and warm
days at the altitudes of about 1500 m above the ground and
3000 m above the ground, respectively (see Georgii and
Miller, 1974; Georgii, 1978). On warm days in summer the
ground concentration of NH3 is considerably higher than
that on the cold days in winter. Sulfur dioxide concentration
in the ABL (atmospheric boundary layer) is higher during
winter than during summer because of the higher
anthropogenic SO2 production caused by enhanced
combustion of fossil fuels in winter. Diurnal and seasonal
variations of CO2 distribution with altitude occur due to
the competition between different phenomena, e.g.
photosynthesis, respiration and thermally driven buoyant
mixing (see Denning et al., 1995). Information about the
evolution of the vertical profile of soluble gases with time
allows calculating fluxes of these gases in an the ABL.
Vertical transport of soluble gases in the ABL is an integral
part of the atmospheric transport of gases and is important
for understanding the global distribution pattern of soluble
trace gases. An improved understanding of the cycle of
soluble gases is also essential for the analysis of global
climate change. Clouds and rains play essential role in
vertical redistribution of SO2 NH3 and other soluble
gases in the atmosphere. Scavenging of soluble gases, e.g.,
Paper No
SO2 NH3 by rain affects the evolution of vertical
distribution of these gases. At the same time the vertical
gradients of the soluble gases concentration in the
atmosphere affect the rate of gas absorption by rain droplets.
Note that the existing models of global transport in the
atmosphere (see, e.g. de Arellano et al., 2004) do not take
into account the influence of rains on biogeochemical cycles
of different gases.
Existence of the vertical temperature distribution in the
atmosphere was discovered in 1749 by A. Wilson (see
Wilson, 1826). Inspired by Wilson, numerous measurements
and modeling of vertical temperature distribution in the
atmosphere (see, e.g. Dines, 1911; Taylor, 1960; Manabe
and Strickler, 1964; Manabe and Wetherald, 1967) revealed
6.5Kkm1 lapse rate. Evolution of the lapse rate during
the last decades is discussed by Trenberth and Smith, (2006).
Vertical temperature profiles in atmosphere during nocturnal
inversions were investigated experimentally and
theoretically, e.g. by Anfossi et al. (1976), Surridge (1986),
Surridge and Swanepoel (1987), Anfossi (1989). Analytical
expressions for nocturnal temperature distribution in the
atmosphere and for decrease of temperature at a ground
during the night were determined by Surridge (1986) and
Anfossi (1989). Influence of vertical distribution of
temperature in the atmosphere on the rate of gas scavenging
by falling rain droplets is explained by a strong nonlinear
dependence of the solubility parameter (Henry's constant)
for aqueous solutions of different gases on temperature (see,
e.g. Reid et al., 1987). Accounting for vertical distributions
of absorbate and temperature in the gaseous phase requires
solution of heat and mass transfer equations which describe
gas absorption by falling rain droplet.
Due to the differences in solubility of gases in liquids,
mass transfer during absorption of soluble gas by droplets in
the presence of inert admixture can be continuousphase
controlled, liquid phase controlled or conjugate.
Continuousphase controlled mass transfer by falling
droplets was discussed by Kaji et al. (1985), Altwicker and
Lindhjem (1988), Waltrop et al. (1991) and Saboni and
Alexandrova (2001). Liquidphase controlled mass transfer
was studied, e.g., by Amokrane and Caussade (1999) and
Chen (2001). Mass transfer controlled by both phases was
analyzed by Walcek and Pruppacher (1984), Alexandrova et
al. (2'" 4), Chen (21i11 14 Elperin and Fominykh (2005),
Elperin et al. (2007ab, 2008, 2009), Kroll et al. (2, i).
Accumulation of the dissolved atmospheric gases in a
falling water droplet during absorption in the presence of
vertical distribution of the absorbate concentration and
temperature in a gaseous phase and circulation of fluid
inside a droplet caused by shear stresses at the interface is
determined by a system of unsteady convective diffusion
and energy conservation equations with timedependent
boundary conditions. Analytical solution of these equations
requires application of rather sophisticated methods (see, e.g.
Bartels and Churchill, 1942 and Ruckenstein, 1967).
The effect of altitudinal distribution of the absorbate in a
gaseous phase on the rate of gas absorption by falling rain
droplets was investigated by Elperin et al. (2I .'). The
suggested approach includes applying the generalized
similarity transformation to a system of transient equations
of convective diffusion and a Duhamel's theorem. Then the
problem reduces to a numerical solution of a linear
7th International Conference on Multiphase Flow
ICMF 2010, Tampa, FL USA, May 30June 4, 2010
convolution Volterra integral equation of the second kind.
In the present study we investigate simultaneous effect
of the absorbate and temperature inhomogenity in a vertical
direction in a gaseous phase on the rate of gas absorption by
falling droplets.
Nomenclature
k
ND =DiC
9y
NT, Ti
ay
m
mo
PeD1 =kRU/Di
PeT =kRU/ai
r
R
t
T
U
x
XblO
Xb20
Xb200
square root of the thermal
diffusivities ratio, a/a2
specific heat, kJ kmole1 K1
molar density at the bulk of fluid,
mol m3
molecular diffusion coefficient,
2 1 s
m2 *s
square root of the diffusivities
ratio, D1/D2
Henry's law constant,
mole litre* atm1
coefficient in Eq. (1)
molar flux density
2 1
mol m s
heat flux density, W m2
distribution coefficient
distribution coefficient at
temperature T20
Peclet number for a moving
droplet
Peclet number for a moving
droplet
radial coordinate, m
droplet radius, m
time, s
temperature, K
translational velocity of a droplet,
1
ms
velocity components, m s1
molar fraction of an absorbate
initial value of molar fraction of
absorbate in a droplet
value of molar fraction of
absorbate in a gas phase at height
H
value of molar fraction of
absorbate in a gas phase on the
ground
7th International Conference on Multiphase Flow
ICMF 2010, Tampa, FL USA, May 30June 4, 2010
Xbl(t)
Xb2(t)
X,(t)= x(t)/mox20
X2 (t)= x2(t)/X20
y
Y = y/R
z
molar fraction of absorbate in a
bulk of a droplet
molar fraction of an absorbate in
a bulk of a gas phase
normalized molar fraction of an
absorbate in a liquid phase
normalized molar fraction of an
absorbate in a gaseous phase
distance from the surface of a
droplet, m
dimensionless distance from the
surface of a droplet
coordinate in a vertical direction,
m
Figure 1: Schematic view of a falling droplet.
dimensionless temperature
molar densities ratio
thickness of a diffusion boundary
layer, m
thickness of a thermal boundary
layer, m
dimensionless thickness of a
boundary layer
thermal
W m1aK
variable
tUk/R
Subscripts
0
conductivity,
similarity variable
angle
dimensionless time
value at the height H in the
atmosphere
liquid phase
gaseous phase
value in the bulk
radial direction
tangential direction
value at gasliquid interface
Description of the model
Consider absorption of a soluble gas from a mixture
containing an inert gas by a moving droplet. At time t = 0 the
droplet begins to absorb gas from the atmosphere.
Distribution of the concentration of the absorbate and
temperature in the gaseous phase in the vertical direction is
assumed to be known (see Fig. 1).
In the analysis we account for the resistance to heat and
mass transfer in both phases and use the following
simplifying assumptions: 1) we employ the approximation
of the infinite dilution of the absorbate in the absorbent; 2)
thicknesses of the diffusion and temperature boundary layers
in both phases are assumed small compared with the
droplet's size; 3) tangential molecular heat and mass transfer
rates along the surface of a spherical droplet are assumed
small compared with a molecular heat and mass transfer
rates in the normal direction; 4) the bulk of a droplet, beyond
the diffusion and temperature boundary layers, is assumed to
be completely mixed and concentration of the absorbate and
temperature are homogeneous in the bulk; 5) the droplet has
a spherical shape; 6) internal circulation inside the droplet
and gas velocity in the vicinity of the droplet have axial
symmetry, 7) solubility parameter (Henry's constant)
depends upon temperature, 8) rain droplet falls without
evaporation.
The assumptions about the circulation inside a droplet
and that the droplet has a spherical shape are valid in the
following ranges of the falling in air water droplet radii,
Reynolds numbers and velocities: 0.1mm
10
fluid flow around a moving droplet showed that at different
Reynolds numbers the tangential fluid velocity component
in the vicinity of a gasliquid interface can be approximated
by the following equation (see, e.g. Pruppacher and Klett
1997, p. 392):
vg = kUsin8, (1)
where coefficient k is equal to 0.04 in the range of the
external flow Reynolds numbers (Re = 2URp2 / 2 ) from
10 to 300 (see, e.g., Pruppacher and Klett 1997, p. 386). The
dependence of the terminal fall velocity of liquid droplets
on their diameter was analyzed by Pruppacher and Klett
(see Pruppacher and Klett 1997, Chapter 10). In this study
we assume that gas absorption does not disturb temperature
distribution in gaseous and liquid phases. At the same time
heat transfer between the atmosphere and a falling droplet
affects the rate of gas absorption/desorption by a falling
droplet. This dependence is explained by a very strong
variation of the solubility parameter (Henry's constant) with
temperature (see, e.g. Reid et al., 1987). Since the
dependence of other thermodynamic parameters on
temperature is by the order of magnitude weaker, we
Paper No
Greek symbols
oi = Tio/Tio
Y = C,/C2
SD,
Ai = 6i /R
Paper No
assume them to be constant. Following the approach
suggested by Ruckenstein (1967) we arrive at the following
system of transient equations of convective diffusion and
energy conservation for the liquid and gaseous phases
which account for convection in radial and tangential
directions:
xi sin xi 2ycos xi D 02X
+Uk += =Di
at R 08 R Dy y2
(2)
Ti Uk sin3 Ti 2ycos8 JTi a 2Ti, (3)
8t R 08 R y a y2
where i = 1, 2. Radial fluid velocity component in Eqs. (2) 
(3) is determined by Eq. (1) and the continuity equation:
vr = 2kcos8 Uy/R Equations (2) (3) are written in a
frame attached to the falling droplet and valid for y << R.
Since the velocity of the droplet fall is known and z = U t,
the vertical coordinatedependent boundary conditions can
be transformed into the timedependent boundary conditions.
The vertical coordinate z is aligned with the direction of the
droplet fall. The initial and boundary conditions to Eqs. (2)
read:
X2 Xb2(t)
X1 X=bl(t)
x1 = x2
ND1 =ND2
The initial and boundary
written as follows:
as y > o,
as y oo,
at y =0,
at y =0.
conditions to Eqs. (3) can be
7th International Conference on Multiphase Flow
ICMF 2010, Tampa, FL USA, May 30June 4, 2010
Method of solution
Since the boundary conditions (4) (5) and (8) (9) to
Eqs. (2) (3) are timedependent, the solution can be found
by combining the similarity transformation method with
Duhamel's theorem. Let us introduce the following
selfsimilar variables (for details see Ruckenstein 1967):
y Y
6 (t,8) AT
y Y
6D, (t) AD,
Since gas absorption (desorption) do not affect
distribution of temperature in both phases, we can solve heat
transfer equation (Eqs. (3) and (8) (11)) independently
from the mass transfer equation. Variables TTi allow us to
obtain solution of a system of partial differential equations
(3) in a following form:
01(Y, 8,T) =
~ ( b2(>bl() erfc Y d
ST bl( l+YTa (AT (, 
(13)
2(Y,a,z )=
a b2) Ta[b2() bi()]er Y d
b 1+YTa [AT2(,Tc)
(14)
Solution of Eqs. (2) with boundary conditions (4) (7)
reads:
X1(Y,, T)=
as y o,
as y  ,
y=0,
at y =0,
where ND, =DiC NT
0 NT
Oi
Oy
y <
coefficient that characterizes the solubility of gases in
liquids, H is Henry's constant (see Seinfeld and Pandis,
2006, p. 288) and Rg is universal gas constant
( Rg = 0.082 atm litre mole1 .K1 ). The equations of
convective diffusion and energy conservation (2) and (3)
describe variations of concentration and temperature near
the gasdroplet interface inside and outside the droplet in a
boundarylayer approximation. The characteristic values of
diffusion Peclet number of the droplet in this study were of
the order of 104 and 102 for the liquid and gaseous
phases, respectively. The characteristic values of
temperature Peclet number of the droplet in this study are of
the order of 102 for both phases. Consequently, the use of
the boundary layer approximation is justified. The solution
of Eqs. (2) (3) can be obtained using the similarity
transformation method which was suggested first by
Ruckenstein (1967).
a 
d0o
1+ m(Os)yD
''er Afc DY)1 d
(15)
X2(Y, ,)=
f X b 2 (G ) 1 +m' "D r f c D
Xi]\. d ......... ., ,1 '*rfcf
O011 +m(,)yD AD,(&,T )
(16)
where YT= CplC/cp2C2 a = /a2 =T,/T10,
A2
D
_ 4 cos(8) 3cos3 4 l ,) ()1
PeD1 sin4(4) 3 l+f(8, T) 3 1+f(8, T)) j
(17)
A2
T
4 cos() cos3 (4)
PeT sin4() 3
f(,T)= tg2 exp(2T) Pe
2T
1f(8,z)
l+f(8,z)
RkU
Di
1(1f(8,T) 31'
3 1+f((8,T)) j
(18)
RkU
PeT
ai
z=tUk/R is a dimensionless time, Y=y/R The
expression for the temperature at the surface of a droplet
reads:
T2 = Tb2(t)
T1 = Tbl(t)
T = T2 at
NT1 = NT2
Paper No
b b2 () bl() (19)
s()= bl(T)+ (19)
l+YTa
The variables bl(z) and Xbl(z) are the unknown
functions of time which can be determined by means of an
integral energy and material balance for the droplet (see, e.g.
UribeRamirez and Korchinsky, 2000 and Elperin et al.,
2007b):
dObi 3 j d81
db_ 3 sin8 d, (20)
dc 2.PeT1 0 YY =0
dXbl 3 a cX1
JdXbl sind. (21)
dc 2.PeD1 0 Y Y=0
Substituting expression for the temperature in the droplet
(Eq. 13) into Eq. (20) yields:
bl(t)=
= 1+ J [b2 b sin d] d
v/.PeT,(l+YTa)o0 0 AT1 
(22)
where bl = Tbl /T10. Substituting expression (15) for the
absorbate concentration in the droplet into Eq. (21) yields:
Xbl(T) = Xb0 +
m0xb20
3 T Xbl() (m(8s ())/mo). Xb2() f sin d0d?
V/" PeD1 oi 1 +m(8,s()) yD j AD,(s, C
(23)
Note that Henry's constant in Eq. (23) is a function of
temperature. The magnitude of temperature at a droplet
interface is determined by Eqs (19) and (22). The functional
dependence of the Henry's law constant vs. temperature
reads (see, e.g. Seinfeld and Pandis, 2006):
lnHA(To) AH*I 1
HA(T) Rg T T (24)
where AH is the enthalpy change due to transfer of
absorbate from the gaseous phase to liquid, Rg is the gas
constant. Time dependence of solute concentration in the
bulk of falling droplets is determined via solving Eqs (19),
(22) (23).
If temperature distribution in the atmosphere is
homogeneous, problem is reduced to solving Eq. (23) with
constant parameter of solubility mo. For the linear vertical
temperature distribution in the atmosphere:
Tb2 = Tb20 + gradTb2 z (25)
where Tb200 temperature at the ground, Tb20 
temperature at the height H from the ground, z is directed
from the cloud to the ground, gradTb2 = (Tb200 Tb20)/H
and z= Ut.
Eq. (23) is a linear convolution Volterra integral
equation of the second kind (Apelblat 2008, Chapter 3). Eq.
(23) can be written in the following form:
t
f(t)= ff(O)K(t,O)d +g(t). (26)
The method of solution of the integral equation (26) is based
The method of solution of the integral equation (26) is based
7th International Conference on Multiphase Flow
ICMF 2010, Tampa, FL USA, May 30June 4, 2010
on approximating integral in Eq. (26) using some quadrature
formula:
b N
JF(=)dS = aiF(i)+RN[F],
a i=1
where Qi e [a,b], i = 1,2,...,N; ai coefficients which are
independent of the function F; RN [F] remainder of the
series after the Nth term. Using a uniform mesh with an
T To
increment h (Ti = To + ih, h TN ) and applying the
N
trapezoidal integration rule yields:
f(0) = g(0),
(1 h.Kii).fi
2
1 i1
h( Kiofo+ Kij fj)+gi,
2j=
where i = 1..., N, fi =f(i.h) gi =g(i h)
Kij =K(ih,j.h).
In order to solve the system of equations (22)(23) we can
represent the equation (26) as a vector equation for the
vector ofM (M = 1, 2) functions f(t). In this case the kernel
K(t, ) is a Mx M matrix and equation (27) can be
viewed as a vector equation. For each i we solved the
M x M set of linear algebraic equations using the Gaussian
elimination method.
Results and Discussion
Calculations of temperatures and concentrations of the
dissolved gas inside water droplets were performed for the
rain droplets with diameter 1.2 mm falling in a
homogeneous and non homogeneous atmosphere containing
a soluble trace gas. It should be noted that this value is equal
to the average diameter of rain droplets of FeingoldLevin
droplet size distribution (Feingold and Levin 1986)
corresponding to low rain intensity of 56 mm/h. Influence
of diurnal variations of CO2 concentration and
temperature distribution in the atmosphere with altitude on
trace gas scavenging by a falling droplet is shown in Figs.
25.
350 360 370 380 390 400
Concentration, C02 (ppm)
Figure 2: Dependence of CO2 concentration in the
atmosphere vs. altitude (1) aircraft measurements Valencia
6:23 (by PerezLanda et al., 2007); (2)(4) approximation of
the measured data; (3) aircraft measurements Valencia 13:03
(by PerezLanda et al., 2007).
Paper No
Vertical profiles of CO2 concentration measured by
PerezLanda et al. (2'" )1 at the heights between 25m and
800m at 06:23 (see curve 1 in Fig. 2) and 13:03 (see curve 3
in Fig. 2) during the growing season over a rice paddy field
in Valencia coastal region, Spain, were approximated by
logarithmic and linear functions (see curves 2 and 4 in Fig.
2) using leastsquares method.
The vertical profiles of the potential temperature
measured by PerezLanda et al., 2007 and atmospheric and
droplet surface temperature calculated using these data are
showed in Figs 3 and 4. The vertical profiles of the
atmospheric temperature were approximated using
polynomial functions of the fifth order. Droplet surface
temperature was calculated using Eq. (19).
600
500
400
300
200
100
294 296 298 300 302
Temperature (K)
Figure 3: Dependence of the potential, atmospheric and
droplet surface temperature in the morning vs. altitude.
S 400
S 300 
200 ,'//
S* '  Atmospheric temperature
Potential temperature, exp
100 PerezLanda et al 2007
 Droplet temperature
296 298 300 302 304 306 308
Temperature (K)
Figure 4: Dependence of the potential, atmospheric and
droplet surface temperature in the afternoon vs. altitude.
Numerical results for C02 concentration inside a falling
droplet are showed in Fig. 5. In our calculations we used the
soluble gas concentration and temperature profiles measured
by PerezLanda et al., 2007 at 06:23 and 13:03. We also
compared the obtained results with our previous calculations
(Fig. 5) performed for uniform temperature distribution in
the atmosphere (see Elperin et al., 2009). The comparison of
the results (see Fig. 5) shows that taking into account the
7th International Conference on Multiphase Flow
ICMF 2010, Tampa, FL USA, May 30June 4, 2010
vertical temperature distribution in the atmosphere changes
the rate of gas absorption by a droplet during its fall.
Inspection of Fig. 5 shows that changing the vertical profiles
of CO2 from the logarithmic early in the morning to the
uniform in the afternoon for uniform temperature
distribution in the atmosphere alters the scenario of the trace
gas absorption by a falling rain droplet. Early in the morning
concentration of the absorbate in the gaseous phase has a
pronounced maximum at the surface and decreases with
altitude. Correspondingly, the droplet absorbs the soluble
gas during all free fall period from a cloud to the ground.
In the afternoon, due to CO2 vertical rectification in
the atmosphere, absorbate concentration in the droplet
attains saturation after a few seconds and at the final stage of
their fall droplets do not absorb soluble trace gases (see Fig.
Time, t (s)
Figure 5: Dependence of the concentration of the
dissolved CO2 gas in the bulk of a falling rain droplet vs. time,
Xbl0 = 0.
Influence of temperature distribution in the atmosphere
on soluble gas scavenging by falling rain droplet is analyzed
via numerical solution of Eqs. (19) and (22)(24).
Dependence of the interfacial temperature of falling rain
droplet vs. altitude in the atmosphere with temperature
gradient equal to 6.5 Kkm1 is showed at Fig. 6.
Droplet temperature, T (K)
Figure 6: Dependence of the interfacial temperature of a
falling rain droplet vs. altitude.
Results of numerical calculations show that if a rain
*, Valencia 6:23
'i
I.
;
fi'
 i/   Droplet surface temperature
 Potential temperature, exp
i PerezLanda et al, 2007
S Atmospheric temperature
 6 23 Valencia
 1303
S 6 23 uniform atmospheric temperature
 13 03 uniform atmospheric temperature
S
Valencia 13:03
 Droplet surface temperature, T
... ....  Atmospheric temperature, T
16 287 288 289 290 291 292 293
1
Paper No
droplet arrives from a region with a lower ambient
temperature to a region with a higher ambient temperature,
droplet's surface temperature at a given height is always
lower than atmospheric temperature at the same height.
When concentration of soluble trace gases in the atmosphere
is constant and temperature distribution is homogeneous,
solute concentration in the droplet attains saturation after a
certain time elapsed, and at the final stage of their fall
droplets do not absorb soluble trace gases, see, e.g. Elperin
and Fominykh (2005).
If temperature in the atmosphere decreases with height,
beginning from some altitude gas absorption is replaced by
gas desorption for a droplet with negligibly small initial
concentration of soluble gas (see Fig. 7). This behavior can
be explained by the dependence of Henry's constant and gas
solubility in liquid on temperature. The higher is the
temperature, the lower is the solubility of trace gas in liquid
and the lower is the concentration of saturation of a trace gas
in a droplet. If a rain droplet arrives from a region with lower
temperature to a region with a higher temperature, after a
while the droplet desorbs the dissolved gas. If the droplet
falling from a cloud is initially saturated, it desorbs gas
during all the fall period.
0.2 0.4 0.6 0.8
Figure 7: Dependence of the normalized concentration of
the dissolved gas in the bulk of a water droplet vs. time for
absorption of ammonia by water in the atmosphere when
temperature in the atmosphere decreases with height with
the rate 6.5 K.km1.
The suggested model of soluble gas scavenging by falling
drops can be incorporated into the existing models of global
transport in the atmosphere to take into account the influence
of rains on biogeochemical cycles of different gases.
Conclusions
In this study we considered conjugate heat and mass
transfer during soluble gas absorption by a falling droplet
with the internal circulation from a mixture containing an
inert gas using approximation of a thin concentration and
temperature boundary layers in the liquid and gaseous
phases and accounting for the absorbate accumulation in the
bulk of the liquid. The bulk of the droplet, beyond the
7th International Conference on Multiphase Flow
ICMF 2010, Tampa, FL USA, May 30June 4, 2010
diffusion and temperature boundary layers, is completely
mixed and concentration of absorbate and temperature is
homogeneous and timedependent in the bulk. The system of
transient partial parabolic differential equations of
convective diffusion and energy conservation in liquid and
gaseous phases with timedependent boundary conditions
has been solved by combining the similarity transformation
method with Duhamel's theorem. The simple form of the
obtained solutions allows using them in the analysis of the
dependence of the rate of heat and mass transfer on different
parameters, e.g., upon the radius of the droplet, diffusion
coefficient, gradient of the absorbate concentration and
gradient of temperature in a gaseous phase etc. The obtained
solution can be also used for validating modeling procedures
for solving more involved problems of gas absorption by
falling liquid droplets.
The results obtained in this study can be summarized as
follows:
1. 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 distributions in a gaseous phase on the
rate of heat and mass transfer.
2. It is shown that the dependence of the radiusaveraged
concentration and temperature vs. time in a falling
droplet is determined by system of linear convolution
Volterra integral equations of the second kind which is
easier to solve numerically than the original system of
partial differential equations.
3. 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.
Neglecting temperature inhomogenity in the atmosphere
described by the adiabatic lapse rate leads to
overestimating the trace gas concentration in a droplet at
the ground by tens of percent. 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. This
behavior can be qualitatively explained by the increase
of trace gas solubility in a droplet with temperature
decrease. At the same time in our model the average
temperature in a droplet at a given altitude is not equal to
the atmospheric gas temperature at the same altitude.
Consequently, the instantaneous concentration of the
dissolved gas in a droplet is not equal to the
concentration of saturation in a liquid corresponding to
the concentration of a trace soluble gas in an atmosphere
at a given height. Therefore the exact quantitative
analysis of the soluble trace gas concentration evolution
in a droplet can be performed only through numerical
solution of a system of integral equations (22)(23).
The developed model can be used for the analysis of
scavenging of hazardous gases in the atmosphere by rain
droplets and can be incorporated into the appropriated
computer codes.
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