numerical analysis of simultaneous heat and mass transfer during absorption of polluted gases by...

Numerical analysis of simultaneous heat and mass transfer during

absorption of polluted gases by cloud droplets

T. Elperin, A. Fominykh and B. Krasovitov

Department of Mechanical EngineeringThe Pearlstone Center for Aeronautical Engineering

Studies Ben-Gurion University of the Negev

P.O.B. 653, Beer Sheva 84105, ISRAEL

Motivation and goals

Description of the model

Results and discussion

Conclusions

Outline of the presentation

Ben-Gurion University of the Negev

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

uniform temperature distribution in both phases was assumed (see e.g., Karamchandani, P., Ray, A. K. and Das, N., 1984);

liquid-phase controlled mass transfer during absorption was investigated when the system consisted of liquid droplet, its vapor and solvable gas (see e.g., Ray A. K., Huckaby J. L. and Shah T., 1987, 1989);

Gas absorption by falling droplets accompanied by subsequent dissociation reaction (see e.g., Baboolal et al. (1981), Walcek and Pruppacher (1984), Alexandrova et al., 2004);

Simultaneous heat and mass transfer during droplet evaporation or growth:

model of physical absorption (Elperin et al., 2005);

model taking into account subsequent dissociation reaction (Elperin et. al, 2007).

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

uniform temperature distribution in both phases was assumed (see e.g., Karamchandani, P., Ray, A. K. and Das, N., 1984);

liquid-phase controlled mass transfer during absorption was investigated when the system consisted of liquid droplet, its vapor and solvable gas (see e.g., Ray A. K., Huckaby J. L. and Shah T., 1987, 1989);

Gas absorption by falling droplets accompanied by subsequent dissociation reaction (see e.g., Baboolal et al. (1981), Walcek and Pruppacher (1984), Alexandrova et al., 2004);

Simultaneous heat and mass transfer during droplet evaporation or growth:

model of physical absorption (Elperin et al., 2005);

model taking into account subsequent dissociation reaction (Elperin et. al, 2007).

Gas absorption by cloud droplets: Scientific background


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

Henry’s Law:

Electro neutrality equation:



Governing equations

1. gaseous phase r > R (t)

022

rrrt

r v

r

YrD

rYr

rY

tr j

jjrj222 v

r

Trk

rTcr

rt

Tcr e

eeprep 222 v

2. liquid phase 0 < r < R (t)

r

Tr

rt

Tr

L

L

L22

r

YrD

rY

tr

L

LLL

LA

A

)(22

(1)

(2)

(3)

(4)

(5)

,1,...,1 Kj

;11 Kj jY

In Eqs. (2)

K

jjjeg MYTRpp

1

Droplet Far

field

Z

X

Y

Gas-liquid interface

R

Gaseous phase

Lmd

ds

Am

)IV(IV SLA

MSY



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-soluble components of the inert admixtures are not absorbed in the liquid

AjjjRJ jj ,1,04 2

(6)

(7)

Taking into account Eq. (7) and using anelastic approximation we can obtain the expression for Stefan velocity:

RrRr

As r

Y

Y

D

r

Y

Y

D LLL 1

1

1

1 11

v (8)

where subscript “1” denotes water vapor species

02

rrr

v

Stefan velocity and droplet vaporization rate



The material balance at the gas-liquid interface yields:

RtRRtd

mds

L ,4 2 v (9)

Then assuming we obtain the following expressions for the rate of change of droplet's radius:

RrRr

A

r

Y

Y

D

r

Y

Y

DR

L

LL 1

1

1

1 11 (10)

L





Rr

s r

Y

Y

D 1

1

1

1v

Rrr

Y

Yρ

DρR

L

1

1

1

1

Rr

s r

Y

Y

D 1

1

1

1v

Rrr

Y

Y

DR

L

1

1

1

1

Rr

A

r

Y

Y

D LLL

11

Rr

A

r

Y

Y

D LL

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



Initial and boundary conditions

The initial conditions for the system of equations (1)–(5) read:

At t = 0, :0 0Rr LL TT 0 LLAA YY 0,

At t = 0, :0Rr rYY jj 0, rTT ee 0,(11)

At the droplet surface:

sjRr

jj Y

r

YD v

(12)

(13)

Rr

Aa

Rr

vRr

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

RReLTT (15)



Initial and boundary conditions

The equilibrium between solvable gaseous and dissolved in liquid species can be expressed using the Henry's law

(16)

where (17)

In the center of the droplet symmetry conditions yields:

(18)

(19)

ASA pHSC *)IV(IV

0

0

rr

T L

00

r

A

r

Y L

At and the ‘soft’ boundary conditions at infinity are imposed 0t r

0

r

j

r

Y0

r

e

r

T

2211*

)IV( 12

H

KK

H

KHH SOS

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 (1)–(5) was solved using the method of lines;

The mesh points are spaced adaptively using the following formula:

Method of numerical solution


,1tR

rx ;0for tRr

,11

tR

rw

;for tRr

;0wx

1,0w 1,0x

;20RtDL

n

i N

ix

1

1,,1 Ni



Average concentration of the absorbed SO2 in the droplet:

Figure 1. Dependence of average aqueoussulfur dioxide molar concentration vs. timefor various values of relative humidity.

Figure 2. Dependence of dimensionless averageaqueous SO2 concentration vs. time for variousinitial sizes of evaporating droplet R0.

relative absorbate concentration is determined as follows:



Figure 3. Droplet surface temperature vs. time(T0 = 278 K, T∞ = 293 K, RH = 70%).

Figure 4. Effect of Stefan flow and heat ofabsorption on droplet surface temperature

(Elperin et al. 2005).

Figure 5. Droplet surface temperature vs. time:1 – model taking into account the equilibriumdissociation reactions; 2 – model of physicalabsorption.



Figure 7. Average concentration of aqueous sulfur species and their sum vs. time ( RH = 101%).

Figure 6. Average concentration of aqueous sulfur species and their sum vs. time ( RH = 70%).

pH is a measure of the acidityor alkalinity of a solution.

Results and discussion: the interrelation between heat and mass transport


Decreases Stefan velocity

Absorption duringdroplet evaporation

Diffusion ofabsorbate

Thermal effect ofabsorption

Reactions ofdissociation

Decreases vapor flux

Increases droplet surface temperature

Increasesvapor flux

Decreases effective Henry’s constant

Decreases dropletsurface temperature

Increases absorbate flux

Increases effective Henry’s constant

Decreases absorbate flux

Increases Stefan velocity

Conclusion


The obtained results show, that the heat and mass transfer rates in water droplet-air-water vapor system at short times are considerably enhanced under the effects of Stefan flow, heat of absorption and dissociation reactions within the droplet.

It was shown that nonlinearity of the dependence of droplet surface temperature vs. time stems from the interaction of different phenomena. Numerical analysis showed that in the case of small concentrations of SO2 in a gaseous phase the effects of Stefan flow and heat of absorption on the droplet surface temperature can be neglected.

The developed model allows to calculate the value of pH vs. time for both evaporating and growing droplets. The performed calculations showed that the dependence of pH increase with the increasing relative humidity (RH).

The performed analysis of gas absorption by liquid droplets accompanied by droplets evaporation and vapor condensation on the surface of liquid droplets can be used in calculations of scavenging of hazardous gases in atmosphere by rain, atmospheric cloud evolution.

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