Simultaneous Heat and Mass

8/3/2019 Simultaneous Heat and Mass

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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


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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


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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 presentationOutline of the presentation


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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


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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


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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


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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)


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? 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)


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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


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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


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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)


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!

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


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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)


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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

!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


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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)


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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 -


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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).


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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


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Fig. 4. Dependence of average aqueous CO2molar concentration vs. time

Fig. 5. Dependence of average aqueous SO2molar concentration vs. time


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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)


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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 Qm).

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 of

absorption on droplet surface temperature.


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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.


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Fig. 15. Dimensionless droplet radius vs. time

R0 = 25 Qm, XSO2 = 0.1 ppm.


R0 = 100Qm, N2/CO2 gaseous mixture.


N2/CO2/H2O gaseous mixture YH2O= 0.011.


N2/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 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)


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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.


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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. 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)


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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.

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Simultaneous Heat and Mass