simultaneous heat and mass
TRANSCRIPT
-
8/3/2019 Simultaneous Heat and Mass
1/29
Simultaneous Heat and Mass Transfer
during Evaporation/Condensation on the
Surface of a Stagnant Droplet in thePresence 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
-
8/3/2019 Simultaneous Heat and Mass
2/29
Laboratory of Turbulent Multiphase Flows
http://www.bgu.ac.il/me/laboratories/tmf/turbulentMultiphaseFlow.html
Head - Professor Tov ElperinPeopleDr. Alexander Eidelman
Dr. Andrew Fominykh
Mr. Ilia Golubev
Dr. Nathan Kleeorin
D
r. Boris KrasovitovMr. Alexander Krein
Mr. Andrew Markovich
Dr. Igor Rogachevskii
Mr. Itsik Sapir-Katiraie
-
8/3/2019 Simultaneous Heat and Mass
3/29
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 presentationOutline of the presentation
-
8/3/2019 Simultaneous Heat and Mass
4/29
A diagram of the mechanism of polluted gases and aerosol
flow through the atmosphere, their in-cloud precipitation
and wet removal.
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
-
8/3/2019 Simultaneous Heat and Mass
5/29
Dispersed-phase controlled isothermal absorption of a pure gas bystagnant 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:Gas absorption by stagnant droplets:Scientific backgroundScientific background
-
8/3/2019 Simultaneous Heat and Mass
6/29
Gas-liquid interface
Vapor phase
Liquid film
Solution
Diffusion of pollutant
molecules throughthe gas
Dissolution into the
liquid at the interface
Diffusion of the
dissolved species
from the interface
into the bulk of theliquid
= pollutant molecule
= pollutant captured in solution
Distancetraveledbythe
pollutedmolecule
Absorption equilibria
OHAOHgA 22
? A AApHOHA ! 2
A
H is the Henrys Law
constant
OHA 2 is the species indissolved state
Henrys Law
-
8/3/2019 Simultaneous Heat and Mass
7/29
Aqueous phase sulfur dioxide/water chemical equilibria
OHSOOHgSO 2222 322 HSOHOHSO
233 SOHHSO
? A2
22
SOH
p
OHSOK
! ? A ? A
? AOHSO
HSOHK
22
3
1
!
? A ? A? A
!
3
23
2
HSO
SOHK
? A ? A ? A ? A ! 233 2 SOHSOOHH
Absorption of SO2 in water results in
OHHOH2
The equilibrium constants for which are
? A? A! OHHKwThe electroneutrality relation reads
(1)
(2)
-
8/3/2019 Simultaneous Heat and Mass
8/29
? A ? A ? A ! 233 2 SOHSOHHuckaby & Ray (1989)
Using the electroneutrality equation (11) and expressions for equilibrium
constants (10) we obtain
? A ? A
? A ? A
? A ? A ? A
? A ? A ? A ? A _
? A 024
4IV
262
IV12IV4
1
2
2121212
22121
22
222
21
2
2
221122
22
12
2
213
2
!
v
K
K
KKKKKKgSOK
KKKKgSOKgSOKgSOKS
gSOKK
K
gSOK
KKKKKKKgSO
SKgSOKKgSO
KKKSK
wwH
HHH
H
w
H
wH
HH
w
tRr!at
where
? A ? A ? A ? A ! 23322IV SOHSOOHSOS
is total dissolved sulfur in solution.
(3)
-
8/3/2019 Simultaneous Heat and Mass
9/29
Gas absorption by stagnant dropletGas absorption by stagnant droplet
Description of the modelDescription of the model
Governing equations
1. gaseous phase r>R (t)
022 !x
x
x
xrr
rtr vV
V
x
x
xx!xxxx rYrD
rYr
rY
tr jjjrj 222 VVV v
x
x
x
x!
x
x
x
x
r
Trk
rTcr
rt
Tcr eeepr
ep 222vV
V
2. liquid phase 0 < r
-
8/3/2019 Simultaneous Heat and Mass
10/29
-
8/3/2019 Simultaneous Heat and Mass
11/29
Stefan velocity and droplet vaporization rate
!!! x
x!
x
x!
Rr
A
Rr
AAsARrA r
YD
r
YDYj
L
LLVVV v
The continuity condition for the radial flux of the absorbate at the dropletsurface reads:
Other non-solvable components of the inert admixtures are not absorbed in the
liquid
AjjjRJ jj {{!! ,1,042
T
(13)
(14)
Taking into account this condition and using Eq. (10) we can obtain the
expression for Stefan velocity:
!!
xx
xx
!
RrRr
As
rY
YD
rY
YD
L
LL 1
1
1
1 11VV
v (15)
where subscript 1 denotes water vapor species
-
8/3/2019 Simultaneous Heat and Mass
12/29
Stefan velocity and droplet vaporization rate
The material balance at the gas-liquid interface yields:
RtRRtd
mds
L ! ,4 2 vVT (16)
Then assuming we obtain the following expression for the
rate of change of droplet's radius:
V V ""L
!!
x
x
x
x
!
RrRr
A
r
Y
Y
D
r
Y
Y
DR
L
L
L 1
1
1
1 11 V
V (17)
-
8/3/2019 Simultaneous Heat and Mass
13/29
Stefan velocity and droplet vaporization rate
!
xx
!
Rr
sr
Y
Y
D 1
1
1
1v
!xx
!
Rr
r
Y
Y
DR
L
1
1
1
1
!
xx
!
Rr
sr
Y
Y
D 1
1
1
1v
!
x
x
!
Rrr
Y
Y
DR
L
1
1
1
1V
V
!x
x
Rr
A
r
Y
Y
D LLL
11V
V
x
x
!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
-
8/3/2019 Simultaneous Heat and Mass
14/29
Initial and boundary conditions
The initial conditions for the system of equations (1)(5) read:
At t = 0, :0 0Rree LL
TT 0! LL
AA YY 0,!At t = 0, :0Rru 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
jj Y
rYD vVV !x
x
!
(19)
For the absorbate boundary condition reads:
!!
x
x!
x
x
Rr
A
Rr
AAsA
r
YD
r
YDY
L
LLVVV v (20)
The droplet temperature can be found from the following equation:
!!!x
x
x
x!
x
x
Rr
Aa
Rr
v
Rr
ee
r
YDL
r
Tk
td
RdL
r
Tk
L
LL
L
LL VV (21)
-
8/3/2019 Simultaneous Heat and Mass
15/29
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
!x
x
!rr
T L 0
0
!x
x
!r
A
r
Y L
At and the softboundary conditions at infinity are imposed0"t gpr
0!x
x
gpr
j
r
Y0!
x
x
gpr
e
r
T
-
8/3/2019 Simultaneous Heat and Mass
16/29
Vapor concentration at the droplet surface andHenrys constant
The vapor concentration (1-st species) at the droplet surface is the functionof temperature T
s(t) and can be determined as follows:
Mp
MTpTYtRY
sss
sss
111 ,,
,1,1 , !!!V
V
where
The functional dependence of the Henry's law constant vs. temperature reads:
(!
0
0 11lnTTR
H
TH
TH
GA
A
g! pp
Fig. 1. Henry's law constant for aqueous
solutions of different solvable gases vs.
temperature.
(26)
(27)
-
8/3/2019 Simultaneous Heat and Mass
17/29
Method of numerical solutionMethod of numerical solution
Spatial coordinate transformation:
The gas-liquid interface is located at
Coordinates x 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
tR
rx ! ;0for tRr
,1
1
!
tR
rw
W ;for tRr"
;0!! wx
? A1,0w ? A1,0x
;20RtDL!X
n
i
N
ix
!
11,,1 ! Ni -
-
8/3/2019 Simultaneous Heat and Mass
18/29
Results and discussionResults and discussion
Fig. 2. Temporal evolution of radius of evaporating water
droplet in dry still air. Solid linepresent model, dashed line
non-conjugate model (Elperin & Krasovitov, 2003), circles
experimental data (Ranz & Marshall,1952).
-
8/3/2019 Simultaneous Heat and Mass
19/29
Fig. 3. Comparison of the numerical results
with the experimental data (Taniguchi &
Asano,1992) and analytical solution.
N.. 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
22n
nnT
T[ !
g
!
dD
tDL!Fo
-
8/3/2019 Simultaneous Heat and Mass
20/29
Fig. 4. Dependence of average aqueous CO2molar concentration vs. time
Fig. 5. Dependence of average aqueous SO2molar concentration vs. time
-
8/3/2019 Simultaneous Heat and Mass
21/29
Typical atmospheric parameters
ReferenceDroplet RadiusCloud-
type/particle type
E. Linacre and B.
Geerts (1999)
4.76.7 Qmstratus
35 Qmcumulus
6 8Qmcumulonimbus
Cooperative
Convective
Precipitation
Experiment (CCOPE)
University of
Wyoming
~20Qmgrowing cumulus
E. Linacre and B.
Geerts (1999)
8Qm 0.5 mmfog
H. R. Pruppacher and
J. D. Klett (1997)
up to 80Qmorographic
~ 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)
-
8/3/2019 Simultaneous Heat and Mass
22/29
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 Qm).
Fig. 8. Dependence of dimensionless
average aqueous SO2 concentration vs.
time (RH = 0%).
-
8/3/2019 Simultaneous Heat and Mass
23/29
Fig. 10. Droplet surface temperature vs. time
(T0 = 274 K, T = 288 K).
Fig. 11. Effect of Stefan flow and heat of
absorption on droplet surface temperature.
-
8/3/2019 Simultaneous Heat and Mass
24/29
Fig. 12. Droplet surface temperature N2/CO2/H2O
gaseous mixture (YH2O= 0.011).
Fig. 13. Droplet surface temperature N2/SO2gaseous mixture.
Fig. 14. Droplet surface temperature N2/NH3gaseous mixture.
-
8/3/2019 Simultaneous Heat and Mass
25/29
Fig. 15. Dimensionless droplet radius vs. time
R0 = 25 Qm, XSO2 = 0.1 ppm.
Fig. 16. Dimensionless droplet radius vs. time
R0 = 100Qm, N2/CO2 gaseous mixture.
Fig. 17. Dimensionless droplet radius vs. time
N2/CO2/H2O gaseous mixture YH2O= 0.011.
Fig. 18. Dimensionless droplet radius vs. time
N2/CO2/H2O gaseous mixture.
-
8/3/2019 Simultaneous Heat and Mass
26/29
Developed model of solvable gas absorption from the mixture with inert gas by falling
droplet (Elperin & Fominykh,Atm. Evironment2005) yields the following Volterra
integral equation of the second kind for the dimensionless mass fraction of an
absorbate in the bulk of a droplet:
(!
TX
PPXU
UP
KTX
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.
)()()()( 22 0 gg! xHxxHtxX AAbb LX
LL DUkRPe ! 0
Lx
)(2 gx
RLL /H!(
RtUk!X
Conjugate Mass Transfer during Gas AbsorptionConjugate Mass Transfer during Gas Absorption
by Falling Liquid Droplet with Internal Circulationby Falling Liquid Droplet with Internal Circulation
(28)
-
8/3/2019 Simultaneous Heat and Mass
27/29
Fig. 19. Dependence of the concentration of the
dissolved gas in the bulk of a water droplet 1-XbVs. time for absorption of CO2by water in the
presence 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 SO2by water in
the presence of inert admixture.
-
8/3/2019 Simultaneous Heat and Mass
28/29
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. Krasovitovand E. R. Shchukin,1991):
g
!sT
T
eeT dTkPe
RJ4
14T
g
g!sT
T
ee
isTm
dTDn
k
cTcJJ
1
,1,1
Where
- dimensionless concentration;
- Peclet number.
nnc 11 !
DT PePePe !!
g
g!E
RUPeT
1D
RUPeD
g!
(29)
(30)
-
8/3/2019 Simultaneous Heat and Mass
29/29
ConclusionsConclusions
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 ofliquid droplet - inert noncondensable and nonabsorbable gas-noncondensable solvable gas.
Droplet evaporation rate, droplet temperature, interfacialabsorbate concentration and the rate of mass transfer during gas
absorption 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 of scavenging of hazardous gases in atmosphere by rain, atmosphericcloud evolution.
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 ofliquid droplet - inert noncondensable and nonabsorbable gas-noncondensable solvable gas.
Droplet evaporation rate, droplet temperature, interfacialabsorbate concentration and the rate of mass transfer during gas
absorption 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 of scavenging of hazardous gases in atmosphere by rain, atmosphericcloud evolution.