simultaneous heat and mass transfer presentation
TRANSCRIPT
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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 ContainingNon-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 NegevP.O.B. 653, Beer Sheva 84105, ISRAEL
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Laboratory of Turbulent Multiphase Flows
http://www.bgu.ac.il/me/laboratories/tmf/turbulentMultiphaseFlow.htmlHead - Professor Tov Elperin
PeopleDr. Alexander Eidelman
Dr. Andrew Fominykh
Mr. Ilia Golubev
Dr. Nathan Kleeorin
Dr. Boris Krasovitov
Mr. Alexander KreinMr. Andrew Markovich
Dr. Igor Rogachevskii
Mr. Itsik Sapir-Katiraie
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Motivation and goalsDescription of the model
Gas absorption by stagnant evaporating/growingdroplets
Gas absorption by moving droplets
Results and discussion: Application for theIn-cloud Scavenging of Polluted Gases
Conclusions
Outline of the presentation
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A diagram of the mechanism of polluted gases and aerosol
flow through the atmosphere, their in-cloud precipitationand wet removal.
NATURAL SOURCES SO2, CO2, CO forestfires, volcanic emissions; NH3 agriculture, wild
animals
ANTHROPOGENICSOURCES SO2, CO2, CO fossilfuels burning (crude oil and
coal), chemical industry; NOx, CO2 boilers,furnaces, internalcombustion and dieselengines; HCl burning of
municipal solid waste(MSW) containing certaintypes of plastics
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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 dropletsuniform temperature distribution in both phases was assumed (seee.g., Karamchandani, P., Ray, A. K. and Das, N., 1984)
liquid-phase controlled mass transfer during absorption wasinvestigated when the system consisted of liquid droplet, its vaporand solvable gas (see e.g., Ray A. K., Huckaby J. L. and Shah T.,1987, 1989)
Simultaneous heat and mass transfer during evaporation/condensation onthe surface of a stagnant droplet in the presence of inert admixturescontaining non-condensable solvable gas (Elperin T., Fominykh A. andKrasovitov B., 2005)
Gas absorption by stagnant droplets:Scientific background
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Gas-liquid interface
Vapor phase
Liquid film
Solution
Diffusion of pollutantmolecules through
the gas
Dissolution into theliquid at the interface
Diffusion of thedissolved speciesfrom the interfaceinto the bulk of theliquid
= pollutant molecule
= pollutant captured in solution
Distancetraveledbyth
epollutedmolecule
Absorption equilibria
OHAOHgA 22
AApHOHA 2
AH is the Henrys Law
constant
OHA 2 is the species indissolved state
Henrys Law
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Aqueous phase sulfur dioxide/water chemical equilibria
OHSOOHgSO 2222 322 HSOHOHSO
233 SOHHSO
2
22
SOH
p
OHSOK
OHSO
HSOHK
22
31
3
23
2
HSO
SOHK
233 2 SOHSOOHH
Absorption of SO2 in water results in
OHHOH2
The equilibrium constants for which are
OHHKw
The electroneutrality relation reads
(1)
(2)
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233 2 SOHSOH
Huckaby & Ray (1989)
Using the electroneutrality equation (11) and expressions for equilibriumconstants (10) we obtain
024
4IV
262
IV12IV4
1
2
21
2
1212
22121
22
222
21
2
2
221122
22
12
2
213
2
K
K
KKKKKKgSOK
KKKKgSOKgSOKgSOKS
gSOKK
K
gSOK
KKKKKKKgSO
SKgSOKKgSO
KKKSK
w
wH
HHH
H
w
H
wH
HH
w
tRrat
where
23322IV SOHSOOHSOS
is total dissolved sulfur in solution.
(3)
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Gas absorption by stagnant droplet
Description of the model
Governing equations1. gaseous phase r>R (t)
022
rr
rtr v
r
Y
rDrYrrYtr
j
jjrj
222
v
r
Trk
rTcr
rt
Tcr eeepr
ep 222v
2. liquid phase 0 < r
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anelastic approximation:
subsonic flow velocities (low Mach number approximation, M
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Stefan velocity and droplet vaporization rate
Rr
A
Rr
AAsARrA r
YD
r
YDYj
L
LL v
The continuity condition for the radial flux of the absorbate at the droplet
surface reads:
Other non-solvable components of the inert admixtures are not absorbed in the
liquid
AjjjRJ jj ,1,042
(13)
(14)
Taking into account this condition and using Eq. (10) we can obtain the
expression for Stefan velocity:
RrRr
As
rY
YD
rY
YD
L
LL 1
1
1
1 11v (15)
where subscript 1 denotes water vapor species
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Stefan velocity and droplet vaporization rate
The material balance at the gas-liquid interface yields:
RtRRtd
mds
L ,4 2 v (16)
Then assuming we obtain the following expression for the
rate of change of droplet's radius:
L
RrRr
A
r
Y
Y
D
r
Y
Y
DR
L
L
L 1
1
1
1 11
(17)
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Stefan velocity and droplet vaporization rate
Rr
srY
YD 1
1
1
1v
Rrr
Y
Y
DR
L
1
1
1
1
Rr
sr
Y
Y
D 1
1
1
1v
Rrr
Y
Y
DR
L
1
1
1
1
Rr
A
rY
YD
L
LL
11
Rr
A
r
Y
Y
DL
L
11
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
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Initial and boundary conditions
The initial conditions for the system of equations (1)(5) read:
At t = 0, :0 0Rr LL
TT 0 LL
AA YY 0,
At t = 0, :0Rr rYY jj 0, rTT ee 0,(18)
At the droplet surface the continuity conditions for the radial flux of non-
solvable gaseous species yield:
sj
Rr
j
j Yr
Y
D v
(19)
For the absorbate boundary condition reads:
Rr
A
Rr
AAsA
r
YD
r
YDY
L
LL v (20)
The droplet temperature can be found from the following equation:
Rr
Aa
Rr
v
Rr
ee
r
YDL
r
Tk
td
RdL
r
Tk
L
LL
L
LL (21)
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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)
AAA pHC
LTTe
0
0
rr
TL
0
0
r
A
r
YL
At and the soft boundary conditions at infinity are imposed0t r
0
r
j
r
Y0
r
e
r
T
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Vapor concentration at the droplet surface andHenrys constant
The vapor concentration (1-st species) at the droplet surface is the function
of temperature Ts(t) and can be determined as follows:
Mp
MTpTYtRY
ssssss
111 ,,,1,1 ,
where
The functional dependence of the Henry's law constant vs. temperature reads:
0
0 11lnTTR
H
TH
TH
GA
A
pp
Fig. 1. Henry's law constant for aqueous
solutions of different solvable gases vs.
temperature.
(26)
(27)
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Method of numerical solution
Spatial coordinate transformation:
The gas-liquid interface is located at
Coordinatesx and w can be treated identically in
numerical calculations;
Time variable transformation:
The system of nonlinear parabolic partial differential equations (4)(8) wassolved using the method of lines;
The mesh points are spaced adaptively using the following formula:
,1
tRrx ;0for tRr
,1
1
tR
rw
;for tRr
;0 wx
1,0w 1,0x
;20RtDL
n
i
N
ix
1
1,,1 Ni
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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).
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Fig. 3. Comparison of the numerical results
with the experimental data (Taniguchi &
Asano, 1992) and analytical solution.
j dddrrrYV
YLL
Ad
A sin1 2
LL
LL
AsA
AA
YY
YY
0,,
0,
Average concentration of absorbed
CO2 in the droplet:
Analytical solution in the case of
aqueous-phase controlled diffusion
in a stagnant non-evaporating
droplet:
Fo4exp161 221
22 nnn
dD
tDLFo
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Fig. 4. Dependence of average aqueous CO2
molar concentration vs. time
Fig. 5. Dependence of average aqueous SO2
molar concentration vs. time
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Typical atmospheric parameters
ReferenceDroplet RadiusCloud-
type/particle type
E. Linacre and B.
Geerts (1999)
4.76.7 mmstratus
35 mmcumulus
68 mmcumulonimbus
Cooperative
ConvectivePrecipitation
Experiment (CCOPE)
University of
Wyoming
~20 mmgrowing cumulus
E. Linacre and B.
Geerts (1999)
8mm0.5 mmfog
H. R. Pruppacher and
J. D. Klett (1997)
up to 80 mmorographic
~ 1.2 mmdrizzle
0.12.0 mmRain drops
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 chemicalreaction (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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Fig. 7. Dependence of dimensionless
average aqueous CO2 concentration vs.
time (RH = 0%).
Fig. 9. Dependence of dimensionless average
aqueous CO2 concentration vs. time
(R0 = 25 mm).
Fig. 8. Dependence of dimensionless
average aqueous SO2 concentration vs.
time (RH = 0%).
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Fig. 10. Droplet surface temperature vs. time(T0 = 274 K, T = 288 K).
Fig. 11. Effect of Stefan flow and heat ofabsorption on droplet surface temperature.
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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. timeN2/CO2/H2O gaseous mixture YH2O= 0.011. Fig. 18. Dimensionless droplet radius vs. timeN2/CO2/H2O gaseous mixture.
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Developed model of solvable gas absorption from the mixture with inert gas by falling
droplet (Elperin & Fominykh,Atm. Evironment2005) yields the following Volterraintegral equation of the second kind for the dimensionless mass fraction of an
absorbate in the bulk of a droplet:
q
q
00),(
sin)(
)1(
31)( dX
DHPeX
LL
bA
b
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.
)()()()( 220 xHxxHtxX AAbb L
LL DUkRPe 0L
x
)(2 x
RLL /
RtUk
Conjugate Mass Transfer during Gas Absorptionby Falling Liquid Droplet with Internal Circulation
(28)
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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 thepresence of inert admixture.
Fig. 20. Dependence of the concentration ofthe 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.
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Heat and mass transfer on the surface of movingdroplet at small Re and Pe numbers
Heat and mass fluxes extracted/delivered from/to the droplet surface (B. Krasovitov
and E. R. Shchukin, 1991):
sT
T
eeT dTkPe
RJ4
14
sT
T
ee
isTm
dTDn
k
cTcJJ
1
,1,1
Where
- dimensionless concentration;
- Peclet number.
n
nc 11
DT PePePe
RUPeT
1
D
RUPeD
(29)
(30)
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Conclusions
In this study we developed a model that takes into account thesimultaneous effect of gas absorption and evaporation
(condensation) for a system consisting of liquid droplet - vapor ofliquid droplet - inert noncondensable and nonabsorbable gas-noncondensable solvable gas.
Droplet evaporation rate, droplet temperature, interfacialabsorbate concentration and the rate of mass transfer during gasabsorption are highly interdependent.
Thermal effect of gas dissolution in a droplet and Stefan flowincreases droplet temperature and mass flux of a volatile speciesfrom the droplet temperature at the initial stage of evaporation.
The obtained results show good agreement with the experimentaldata .
The performed analysis of gas absorption by liquid dropletsaccompanied by droplets evaporation and vapor condensation onthe surface of liquid droplets can be used in calculations ofscavenging of hazardous gases in atmosphere by rain, atmosphericcloud evolution.