Numerical Analysis of Simultaneous Heat and Mass

7/23/2019 Numerical Analysis of Simultaneous Heat and Mass

1/16

Numerical analysis of simultaneousheat and mass transfer during

absorption of polluted gases bycloud droplets

T. Elperin, A. Fominykh and B. Krasovitov

Department of Mechanical EngineeringThe Pearlstone Center for Aeronautical Engineering

StudiesBen-Gurion Universit of the !egev

P"#"B" $%&' Beer Sheva ()*+%' ,SAE.


2/16

Motivation and goals

Description of the model

Results and discussion

Conclusions

Outline of the presentation

Ben-Gurion University of the Negev


3/16

Effect of vapor condensation at the surface of stagnant droplets on therate of mass transfer during gas absorption by growing droplets:

uniform temperature distribution in both phases was assumed (seee.g., Karamchandani, P., Ray, A. K. and as, !., "#$%&'

liuid)phase controlled mass transfer during absorption wasinvestigated when the system consisted of liuid droplet, its vapor

and solvable gas (see e.g., Ray A. K., *uc+aby . -. and hah /.,"#$0, "#$#&'

1as absorption by falling droplets accompanied by subseuentdissociation reaction (see e.g., 2aboolal et al. ("#$"&, 3alce+ andPruppacher ("#$%&, Ale4androva et al., 566%&'

imultaneous heat and mass transfer during droplet evaporation orgrowth:

model of physical absorption (Elperin et al., 5667&'

model ta+ing into account subseuent dissociation reaction (Elperinet. al, 5660&.

Gas absorption by cloud droplets: Scientific background



4/16

Gas-liquid

interface

Droplet

Absorption equilibria


=pollutant molecule

=pollutant captured in solution

Air

SO2

Aqueous phase sulfur dioxide/water

chemical equilibria

is the species in dissolved state

*enry8s -aw:

Electro neutrality euation:


5/16



Governing equations

1. gaseous phaser>R (t)

( ) 022 =

+

rrrt

r v

( ) ( )

=

+

r

YrD

rYr

rY

tr

jjjrj

222 v

( )

=

+

r

Trk

rTcr

rt

Tcr eeepr

ep 222v

2. liquid phase0 < r


6/16



( )

+ ===

=

=Rr

A

Rr

AAsARrA r

YD

r

YDYj

L

LL v

#he $o%iui%& $odi%io 'or %he radial 'lu o' %he asora%e a% %he drople%sur'a$e reads*

O%her o+solule $opoe%s o' %he ier% adi%ures are o% asored i %he

liquid

AjjjRJ jj == ,1,042

(-)

()

#a/ig i%o a$$ou% !q. () ad usig aelas%i$ approia%io

e $a o%ai %he epressio 'or S%e'a elo$i%&*

( )

( )

( )+ ==

=

RrRr

As

r

Y

Y

D

r

Y

Y

D L

LL 1

1

1

1 11

v ()

here sus$rip% 91deo%es a%er apor spe$ies

( ) 02 =

rrr

v

Stefan velocity and droplet vaporization rate


7/16



#he a%erial ala$e a% %he gas+liquid i%er'a$e &ields*

( )( )RtRRtd

mds

L = ,4 2 v ()

#he assuig e o%ai %he 'olloig epressios 'or %hera%e o' $hage o' drople%s radius*

( )

( )

( )+ ==

+

=

RrRr

A

r

Y

Y

D

r

Y

Y

DR

L

L

L 1

1

1

1 11

(10)

>>L



8/16




( )+=

=

Rr

sr

Y

Y

D 1

1

1

1v

( ) +=

=

Rrr

Y

Y

DR

L

1

1

1

1

( ) +=

= Rrs rY

Y

D 1

1

1

1v

( )+=

=

Rrr

Y

Y

DR

L

1

1

1

1

( )

( )

=

Rr

A

r

Y

Y

D L

LL

11

( )

( )

+

=Rr

A

r

Y

Y

D L

L

11

I %he $ase he all o' %he ier%

adi%ures are o% asored i

liquid %he epressios 'or S%e'a

elo$i%& ad ra%e o' $hage o'

drople% radius read


9/16



Initial and boundary conditions

#he ii%ial $odi%ios 'or %he s&s%e o' equa%ios (1)(5) read*

A% % 6 0, *0 0Rr ( ) ( )LL TT 0=

( ) ( )LLAA YY 0,=

A% % 6 0, *0Rr ( )rYY jj 0,= ( )rTT ee 0,=(11)

A% %he drople% sur'a$e*

sj

Rr

jj Y

r

YD v =

+=

(12)

(13)

( ) ( )

+ ===

=+

Rr

Aa

Rr

v

Rr

ee

r

YDL

r

Tk

td

RdL

r

Tk

L

LL

L

LL (14)

( )

+ ==

=

Rr

A

Rr

AAsA

r

YD

r

YDY

L

LL v

( )

+=

RRe

LTT (15)


10/16



Initial and boundary conditions

#he equiliriu e%ee solale gaseous ad dissoled i liquid spe$ies$a e epressed usig %he 7er&s la

(1-)

here

(1)

I %he $e%er o' %he drople% s&e%r& $odi%ios &ields*

(1)

(1)

( )[ ] ASA pHSC8

)I"(I" ==

( )

0

0

=

=rr

T L ( )

0

0

=

=r

A

r

Y L

A% ad %he ;so'%8oudar& $odi%ios a% i'ii%& are iposed0>t r

0=

r

j

r

Y0=

r

e

r

T

[ ] [ ]

++= ++ 22118

)I"( 12H

KK

H

KHH SOS


11/16

Spa%ial $oordia%e %ras'ora%io*

#he gas+liquid i%er'a$e is lo$a%ed a% 9oordia%esxad w$a e %rea%ed ide%i$all& i

ueri$al $al$ula%ios;

#ie ariale %ras'ora%io*

#he s&s%e o' oliear paraoli$ par%ial di''ere%ial equa%ios (1)(5) assoled usig %he e%hod o' lies;

#he esh poi%s are spa$ed adap%iel& usig %he 'olloig 'orula*

Method of numerical solution


( ),1

tRrx = ( );0'or tRr


12/16



Aerage $o$e%ra%io o' %he

asored SO2

i %he drople%*

Figure 1.:epede$e o' aerage aqueous

sul'ur dioide olar $o$e%ra%io s. %ie

'or arious alues o' rela%ie huidi%&.

Figure 2.:epede$e o' diesioless aerage

aqueous SO2 $o$e%ra%io s. %ie 'or arious

ii%ial sies o' eapora%ig drople%R0.

rela%ie asora%e $o$e%ra%io

is de%eried as 'ollos*


13/16



Figure 3.:rople% sur'a$e %epera%ure s. %ie

(#06 2 , #=6 23 , 7 6 0?).

Figure 4.!''e$% o' S%e'a 'lo ad hea% o'

asorp%io o drople% sur'a$e %epera%ure

(!lperi e% al. 2005).

Figure 5.:rople% sur'a$e %epera%ure s. %ie*

1 odel %a/ig i%o a$$ou% %he equiliriudisso$ia%io rea$%ios; 2 odel o' ph&si$al

asorp%io.


14/16



Figure 7.Aerage $o$e%ra%io o' aqueous

sul'ur spe$ies ad %heir su s. %ie ( 7 6101?).

Figure 6.Aerage $o$e%ra%io o' aqueous

sul'ur spe$ies ad %heir su s. %ie ( 7 6

0?).

pHis a easure o' %he a$idi%&

or al/alii%& o' a solu%io.


15/16

Results and discussion: the interrelation beteenheat and mass transport


Decreases Stefan

velocity

Absorption during

droplet evaporation

Diffusion of

absorbate

hermal effect of

absorption

Reactions of

dissociation

Decreases vapor flu!

"ncreases droplet surface

temperature

"ncreases

vapor flu!

Decreases effective

#enry$s constant

Decreases droplet

surface temperature

"ncreases absorbate

flu!

"ncreases effective

#enry$s constant

Decreases absorbate

flu!

"ncreases Stefan

velocity


16/16

!onclusion


#he o%aied resul%s sho, %ha% %he hea% ad ass %ras'er ra%es i a%erdrople%+air+a%er apor s&s%e a% shor% %ies are $osideral& eha$eduder %he e''e$%s o' S%e'a 'lo, hea% o' asorp%io ad disso$ia%io rea$%iosi%hi %he drople%.

I% as sho %ha% olieari%& o' %he depede$e o' drople% sur'a$e%epera%ure s. %ie s%es 'ro %he i%era$%io o' di''ere% pheoea.

@ueri$al aal&sis shoed %ha% i %he $ase o' sall $o$e%ra%ios o' SO2ia gaseous phase %he e''e$%s o' S%e'a 'lo ad hea% o' asorp%io o %hedrople% sur'a$e %epera%ure $a e egle$%ed.

#he deeloped odel allos %o $al$ula%e %he alue o' p7 s. %ie 'or o%heapora%ig ad groig drople%s. #he per'ored $al$ula%ios shoed %ha%%he depede$e o' p7 i$rease i%h %he i$reasig rela%ie huidi%& (7).

#he per'ored aal&sis o' gas asorp%io & liquid drople%s a$$opaied &drople%s eapora%io ad apor $odesa%io o %he sur'a$e o' liquid drople%s$a e used i $al$ula%ios o' s$aegig o' haardous gases i a%osphere& rai, a%ospheri$ $loud eolu%io.

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Numerical Analysis of Simultaneous Heat and Mass