Module 6 Air Water Interface Mass Transfer Discussion In this module you learned about air-water exchange (mass interphase transfer). For the module discussion, pose a question that you have about one of the topics. Use this as an opportunity to extend your knowledge or to clarify a concept. You will be assessed on a) how well you are able to put the question in context, b) your answer to this question, and c) your replies to others.Make an Initial Post: Provide the question and provide a satisfactory answer citing relevant literature. This should not be a simple yes or no question. Initial posts should be at least 300 words.Resource Link CONTAMINANT FATE AND TRANSPORT

AIR-WATER EXCHANGE (Interface Mass Transfer)

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AIR-WATER EXCHANGE (Interphase Mass Transfer)

In this section we will consider transfer of contaminants between air and

water when the two phases are not in equilibrium (e.g. shortly after a

catastrophic event).

A simplified description of the mechanisms involved in air-water transport of

a contaminant is depicted in Figure 1.

Accordingly the vertical transport of contaminants proceeds through four

distinct layers:

turbulent air bulk

quiescent air layer (~1 mm thickness)

quiescent water layer (~0.1 mm thickness)

turbulent water bulk

Vertical contaminant transport proceeds at fast rates in air and water bulk and

at much slower rates through the two quiescent air and water thin films.

Consequently, vertical transport rate of contaminant transport is controlled by

the transport rates through the thin air/water films.

Based on conceptual simplifications made by various researchers a number of

mass transfer theories are found in the literature.

8.1

Some of the most widely used are presented in the following sections.

STAGNANT FILM MODEL(1904)1

Originally applied to turbulent fluid flow past a solid surface.

The theory postulates the existence of a viscous fluid layer or film in the

proximity of the solid surface where fluid velocities are equal to zero.

1

Nernst W. Z.. Phys. Chem., 47, 52, 1904.

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Advective

Transport

AIR BULK

Turbulent

Transport

Diffusive

Transport

AIR

BOUNDARY LAYER

AIR-WATER

Interphase

Transport

WATER

BOUNDARY LAYER

WATER BULK

Advective

Transport

Diffusive

Transport

~1 mm

~0.1 mm

Turbulent

Transport

Figure 1: Air-Water Transport Mechanisms

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Beyond the stagnant film turbulent flow conditions exist and mixing is very

rapid. Thus the overall mass transfer is limited by the rate of molecular

diffusion within the stagnant film (see Fig. 2).

z

Turbulent Flow

Fluid Bulk

Cf

Stagnant Layer

z

Cs

x

Solid Surface

Figure 2: Stagnant Film Model for Interface Mass Transfer

The diffusional flux within the stagnant film along the z-direction is given by

equation:

F D

C f C s

dC

D

dz

z

[1]

where C f molar conc. of a contaminant in the fluid bulk, mol/m3

C s molar conc. of a contaminant on the solid surface, mol/m3

Define k which represents the partial transfer velocity as:

k

D

z

[2]

where k interfacial mass transfer coefficient, m/s.

And combining Eqs. 1 and 2:

F k C s C f

[3]

CONTAMINANT FATE AND TRANSPORT

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Consequently, the dimensionless Sherwood number is defined as:

N Sh

k L Interfacial Mass Transfer Impedance

D

Molecular Diffusion Impedance

[4]

where L characteristic length, replacing the fictitious film thickness z

The Sherwood number depends on the Reynolds and Schmidt numbers and

system geometry:

N Sh f N Re , N Sc , geometry

N Re

N Sc

8.2

LV

[5]

D

TWO-FILM THEORY (1923)2

This model was first applied to gas-liquid absorption by Whitman and

essentially considers the presence of two stagnant films (one gas, one liquid)

in contact with each other.

Accordingly, gas-liquid interface mass transfer is limited due to diffusion

through these two films.

Figure 3 is a conceptual schematic of the two-film model.

The model is based on two important assumptions:

the contaminant does not undergo any transformations within the

stagnant layers

the concentrations at layer boundaries are at steady state

The flux of molecules across the air layer is given by Fick’s Law of molecular

diffusion:

2

Whitman W. G., “The Two-film Theory of Gas Absorption”, Chem. Metal. Eng., 29, 146-148, 1923

CONTAMINANT FATE AND TRANSPORT

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Completely Mixed Air

za

Stagnant Air Film

Fa

Henry’s Law Equilibrium

at Water-Air Interface

0

Stagnant Water Film

FW

-zw

Completely Mixed Water

Ca

Cw/a

Ca/w

Cw

Concentration, mol/l

Figure 3: Conceptual Sketch of the Two-Film Theory

CONTAMINANT FATE AND TRANSPORT

AIR-WATER EXCHANGE (Interface Mass Transfer)

Fa Da

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C Ca / w

dC

Da a

za

dz

[6]

where C a contaminant concentration in the air bulk, mol/m3

C a / w contaminant concentration in the air-water interface, mol/m3

Da contaminant diffusivity in air, m2/s

Similarly the contaminant flux across the water stagnant layer is:

Fw Dw

C

Cw

dC

Dw w / a

dz

zw

[7]

where C w contaminant concentration in the water bulk, mol/m3

C w / a contaminant concentration in the water-air interface, mol/m3

Dw contaminant diffusivity in water, m2/s

At steady state the contaminant flux through each layer must be the same,

thus:

F Dw

C

w/ a

Cw

zw

Da

C

a

Ca/ w

za

[8]

At the air-water interface equilibrium is established and Henry’s Law is

applicable:

K H

Ca/w

Cw/ a

[9]

Substitution of Eq. 9 into Eq. 8 and solution of the latter for C w / a gives:

Dw C w C w / a Da K H C w / a C a

zw

za

Cw/a

Dw C Da C

z w w

z a a

Dw Da K H

z

z

w

a

Substitution of Eq. 10 into Eq. 7 yields:

[10]

CONTAMINANT FATE AND TRANSPORT

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1

C C a

F

z z w K H

a

w

Dw Da K H

[11]

Define the partial transfer velocities for each of the two stagnant layers:

kw

Dw

zw

[12]

D

ka a

za

where k w water film mass transfer coefficient, m/s

k a air film mass transfer coefficient, m/s

Substitute Eq. 12 into 11 to get:

F

1

1

1

k w k a K H

Ca

Cw

K H

[13]

Define

1

1

1

k k w k a K H

[14]

where k overall mass transfer coefficient, m/s

Evidently Eq. 14 is analogous to the equation describing the total resistance of

two resistors connected in series. Thus it can be deduced that the film mass

transfer coefficients ( k w , k a ) represent a measure of the resistance of each

individual stagnant film to mass transfer and that the overall mass transfer

coefficient ( k ) represents the total system resistance to mass transfer.

Combination of Eqs. 13,and 14 yields:

CONTAMINANT FATE AND TRANSPORT

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C

F k C w a

K H

[15]

The value of the overal mass transfer coefficient is affected by the values of

the film transfer coefficients. Two extreme situations can be distinguished:

Mass transfer is dominanted by the water boundary layer

k w k a K H k k w

[16]

Mass transfer is dominated by the air boundary layer

k w k a K H k k a K H

8.3

[17]

PENETRATION THEORY (1935)3

Liquid elements or “packets” are transported to the air-water interface, where

they remain for some time before they return to the bulk liquid phase.

Mass transfer of gas into the liquid element while it resides at the interface is

shown in Fig. 4 where a liquid element is falling along a vertical wall of

dimensions L (length) by W (width)

A mass balance over a differential volume (dx, dy, dz) gives:

F

F

dx dy W +

dy W dx = 0

x

y

Division by the element volume dx dy W gives:

F F

+

=0

x y

3

[19]

Subsituting the advective and diffusive flux terms in Eq. 19 we obtain:

Vmax

[18]

C

2C

D 2

x

y

[20]

Eq. 20 is another form of Fick’s 2nd Law, if t x Vmax is substituted.

Higbie R., “The Rate of Adsorption of a Pure Gas into a Still Liquid during Short Periods of Exposure”, Trans.

Am. Inst. Chem. Eng., 35, 365-389, 1935.

CONTAMINANT FATE AND TRANSPORT

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W

C0 at y =

Cs at y=0

Water Film

L

Gas Bulk

Vx

z

Diffusion

y

Advection

x

Figure 4: Conceptual Schematic of the Penetration Model

CONTAMINANT FATE AND TRANSPORT

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The above equation can be solved subject to the following initial and

boundary conditions:

C C 0 at y > 0, x = 0

Initial Condition

Boundary Conditions

C = C 0 at y , x > 0

C = C s at y = 0, x > 0

The solution by taking Laplace transforms is given by Welty et al.4

4 Dx 0 .5

C C0

erfc y

Cs C0

V

max

[21]

According to Fick’s 1st Law, the flux of gas into the liquid stream at the airwater interface (y=0) is:

F D

C

y

[22]

y 0

The derivative C y obtained from Eq. 21 can be substituted into Eq. 22 to

yield:

DVmax

F

x

0 .5

C

s

C0

The average interfacial flux over wall width W and length L is:

1 LW

F

F dz dx

WL 0 0

4

[24]

Substituion of Eq. 23 into 24 and integration of the latter gives:

DVmax

F 2

L

[23]

0 .5

C

s

C0

[25]

Thus the mass transfer coefficient is:

Welty J. R., Wicks C. E., Wilson R. E., Fundamentals of Momentum, Heat, and Mass Transfer, 3rd Ed., Wiley

Interscience, pp. 558-560, 1984.

CONTAMINANT FATE AND TRANSPORT

AIR-WATER EXCHANGE (Interface Mass Transfer)

D

k 2

t

where t

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11

0 .5

[26]

L

contact time between air and water.

Vmax

Comparing Eqs. 2 and 26 we can conclude that the dependence of k on

molecular diffusivity is lessened in the penetration model (D is raised to the

0.5 power) compared to the stagnant film theory (D is raised to the 1 power).

8.4

SURFACE RENEWAL THEORY (1951)5

The theory is based on the continuous turnover of air and water “packets”

with their associated organic load at the air-water interface.

After some turnover event and upon arrival at the interface, water and air

“packets” come to equilibrium with each other as described by Henry’s Law

(Eq. 9):

C a / w K H C w / a

[9]

The theory is graphically depicted in Fig. 5

Immediately after exposure of new “packets” across the interface the

chemical starts moving from the water into the air. At time t a total amount of

contaminant fw (kg/m2) has left the water and an equal amount of contaminant

fa appears on the air side of the interface:

fw fa

[27]

The penetration depth is given by the Einstein-Smoluchowski Eq.:

x 2 dt

5

Thus fw and fa can be approximated by the product of xC:

Danckwerts P. V., “Significance of Liquid-Film Coefficients in Gas Absorption”, Ind. Eng. Chem., 43, 1460-1467,

1951.

CONTAMINANT FATE AND TRANSPORT

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z

AIR

Vertical

Distance

gain into air

INTERFACE

fa

Henry’s law equilibrium

reached immediately after

contact of air-water “packets”

z=0

fw

loss from

water

Ca

Cw/a

Ca/w

WATER

Cw

C

Concentration

Figure 5: Surface Renewal Model for Air-Water Mass Transfer

CONTAMINANT FATE AND TRANSPORT

AIR-WATER EXCHANGE (Interface Mass Transfer)

f a Cons tan t C a / w C a Da t

f w Cons tan t C w C w / a Da t

C w Dw C a Da

[29]

Dw K H Da

Inserting Eq. 29 into the second Eq. 28 we get:

f w f a cons tan t

[28]

Solving the above equations for Cw/a:

Cw/ a

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Ca

Cw

K H

1

1

Dw t K H Da t

[30]

Division of fw by time t yields the flux. Note that sometimes the use of r, the

surface renewal rate (equal to 1/t) is preferred:

f

1

F w r t w cons tan t

t

1

1

rDw K H rDa

Ca

Cw

[31]

K H

The surface renewal rate r and the contact time t are idealized model

parameters to be experimentally determined. They can be set equal to 1 or by

decoupling the surface renewal processes on either side of the interface two

surface renewal rates (one for each side) rw and ra can be introduced.

1

1

1

It is apparent that Eq. 31 can be

By defining: k

Dw t K H Da t

written:

CONTAMINANT FATE AND TRANSPORT

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1

1

Ca

1

f w f a cons tan t

Cw

K H

k w k a K H

[32]

C

f w f a cons tan t k C w a

K H

where k w

Dw

, ka

t

Da

t

Summary:

Comparing the three mass transfer theories we see that the essential

difference lies in the definition of mass transfer coefficient see Table 1.

Table 1: Comparison of Mass Transfer Coefficients Derived from Different Mass

Transfer Theories

Theory

Coefficient

D

z

Film

k

Penetration

D

k 2

t

Surface Renewal

k D / t

0 .5

0 .5

Apparently k D a where the exponent a varies between 0.5 to 1.0

The overall volumetric mass transfer coefficient (KLa) for a compound can be

related to that of a known compound by the the following equation:

K a

L

j

Dj

K L a i

Di

[33]

where = system specific factor determined experimentally (for many

VOC’s = 0.6).

CONTAMINANT FATE AND TRANSPORT

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EXAMPLE 1

A municipal wastewater treated by an activated sludge process using surface

aerators. The following data are available on the operation of the aeration tank:

Flowrate Q = 160 m3/d

HRT, t V Q 20 hr

O2 mass transfer requirement per unit volume of wastewater M/V = 400 g/m3

Oxygen transfer rating (field conditions) for mechanical aerators = 3 kg O2/kWh.

The wastewater is known to contain TCE and air pollution regulations require an

evaluation of TCE release due to volatilization from the aeration tank. Since TCE

analysis requires sophisticated and expensive equipment not available in WWTP’s.

an oxygen transfer study was conducted in the setup shown below.

The water was initially stripped of O2 and then measurements of DO were made

with time at a fixed mixer speed. The vessel volume was 8.0 L, the power input Pw

= 2 W and the O2 saturation concentration Cs = 9.0 mg/l. The following data were

collected:

Time, s

0

50

100

200

400

600

800

1200

DO, mg/l

0

1.6

3.1

5.0

7.3

8.3

8.8

8.9

Estimate the percent removal of TCE in the aerobic reactor.

SOLUTION

1. The overall volumetric mass transfer coefficient KLa must be first determined:

KL a k a

where a = the interface area. Assuming complete mixing we have:

dC

a F K L a C s C

dt

Integration of this equation gives:

CONTAMINANT FATE AND TRANSPORT

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RPM Control

Watt Meter

DO Meter

Variable Speed

Motor

Baffles

Impeller

DO/Tempera

ture

Air Flow

Meter

Diffuser

Mixing Tank

V=8.0 L

Air Compressor

Figure 6: Experimental Setup Used for Determination of Mass Transfer

Coefficients

CONTAMINANT FATE AND TRANSPORT

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t

dC

K L a 0 dt

Cs C

C

0

Cs C

KL a t

Cs

ln

2. The data are tabulated in Table 2 and are plotted in Fig. 7. From Fig. 7 the

slope of the regression line gives: K L a 0.004 s -1

3. Eq. 33 can be used to estimate KLa for TCE.

K a

L

TCE

K L a O

2

DTCE

DO

0 .6

2

Diffusivities for O2 and TCE are obtained for literature:

DTCE 0.86 10 6 cm 2 s

DO 1.88 10 6 cm 2 s

2

Substitution of the known values in the above equation gives:

K a

L

TCE

0.86 10 6

0.004

1.88 10 6

0 .6

0.00156 s -1

4. The above value of (KLa)TCE has been derived from laboratory data. The

following equation can be used to convert it to field conditions:

K a

L

field

K L a lab

P

P

w

w

V field

V lab

From the information provided for the lab scale experiment we have:

P

w

V

2W

250 W m 3

3

3

8 10 m

The (Pw/V)field can be estimated from the O2 requirements and the aerator transfer

rating:

CONTAMINANT FATE AND TRANSPORT

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Table 2: Data for Solution of Example 1

Time, s

0

50

100

200

400

600

800

1200

ln C s C ) C s

Conc., mg/l

0

1.6

3.1

5

7.3

8.3

8.8

8.9

800

1000

ln(Cs-C)/Cs

Time, s

C, mg/l

ln[(Cs-C)/Cs]

0.000

0

0

0.000

50 0

1.6

-0.500

200

400 -0.196600

100

3.1

-0.422

-1.000

200

5

-0.811

-1.500

400

7.3

-1.667

-2.000

600

8.3

-2.554

-2.500

800

8.8

-3.807

-3.000

1200

8.9

-4.500

-3.500

-4.000

-4.500

-5.000

0.000

-0.196

-0.422

-0.811

-1.667

-2.554

-3.807

-4.500

Time, s

Figure 7: Plot of Data for Example 1.

1200

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g 1

M 1

3

O 2 Transfer Rate = M O 400 3

20 g m hr

V t

m 20 hr

2

P

w

V field

20 10 3 kg m 3 hr

6.7 W m 3

3 kg kW hr

5. The percent removal of TCE in the aeration tank can be obtained by means of a

mass balance assuming a completely mixed flow regime:

Q C IN Q C OUT K L a C air C OUT V 0

The concentration of TCE in the atmospheric air above the reactor should be

negligible, thus Cair 0. After this simplification and rearrangement of the above

equation we have:

K L aV Q

C

R 1 OUT 100

100

C IN

1 K L aV Q

4 .2 10 5 s 1 20hr 3600s / hr

R

100 751

.%

1 4 .2 10 5 s 1 20hr 3600s / hr

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AIR-WATER EXCHANGE (Interface Mass Transfer)

8.5

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AIR-WATER EXCHANGE IN NATURAL SYSTEMS (ka)

Air-water exchange in natural systems is different from that in laboratory

settings. The physical properties of boundary layers depend on the size of the

wind shear stress at the water surface.

For most applications wind speeds are from different heights are converted to

speeds at a reference height (normally 10 m above the water surface) by the

equation:

ln h 8.1

Vh

V

10.4 10

where h = height above water surface, m

Vh = wind speed at height h, m/s

V10 = wind speed at height 10 m, m/s

8.5.1 AIR MASS TRANSFER COEFFICIENT

Let us first consider water evaporation from a natural water body. Since the

water boundary layer is saturated in water, it can be deduced that kw ,

thus the flux of water vapor is dominated by air boundary layer resistances or

k kaKH’. Thus the flu equation becomes:

Fw k a K H C w C a

Define the relative humidity from Henry’s Law:

H

Ca

C

sata

K H C w C a

[41]

Combining Eqs. 40 and 41 we obtain:

ka

[40]

Fw

F

F

sat w sat sat w

K H C w C a C a HC a C a 1 H

[42]

The value of ka varies in the range of 0.3 to 3.0 cm/s and is affected

significantly by wind speeds.

CONTAMINANT FATE AND TRANSPORT

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Empirical equations used …

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