Gas transfer rates

If either phase concentration can not be predicted

by Henry's law then there will be a transfer of

mass across the interface until equilibrium is

reached.

The mechanisms and rate expressions for this

transfer process have been conceptualized in a

variety of ways so that quantitative descriptions

are possible.

One common conceptualization method is the film

theory.

Film Theory Developed by Nernst (1904)

• The simplest conceptualization of the gas-

liquid transfer process

• Near the interface, there exists a stagnant film.

• This stagnant film is hypothetical since we

really don't know the details of the velocity

profile near the interface.

• In this film, bulk liquid is not moving.

Therefore, transport is governed essentially

by molecular diffusion.

• Fick's law describes flux through the film.

Molecular diffusion

A “random” and spontaneous mixing process that moves the

“solute” from regions of high concentration to regions of low

concentration. Molecular diffusion is due to concentration gradients

and the random motion of molecules.

calm environment

Molecular diffusion

Rates of simple one-dimensional microscale mass transfer

processes in homogeneous fluids or solids are expressed in

terms of Fick’s first law of diffusion.

According to “Fick’s first law”, the flux of component "c" across a

chemical front per unit time is directly proportional to the size of

the spatial gradient in concentration and a constant of

proportionality (the diffusion constant – D )

Area that is perpendicular

to the flow.

Substance moves due to

concentration differences.

(cm2/sec) specific to solute

Molecular diffusion

Therefore, flux through the film is

CJ D

X

Typical units for flux are

2

mol

cm sec 2

mg

cm secor

If the thickness of the stagnant film is given by

dn then the gradient can be approximated by:

C C Cb i

X n

d

Cb and Ci are concentrations in the bulk and at

the interface, respectively.

• Assume there are no reactions in the stagnant

film.

• Therefore, no accumulation in the film.

• Assume that D = constant

At steady-state:

(C C )b iJ Dn

d

What is the

value of Ci?

• Ci is calculated by assuming that equilibrium is

attained instantly at the interface.

• Nernst’s model assumes that the other phase

doesn't have a "film".

CgCi

Hc

(if the film side is liquid and the

opposite side is the gas phase).

Two film model

• Nernst’s model assumes that the other phase

doesn't have a "film".

• This theory is the most widely accepted

model.

• It makes it easier to visualize the concepts.

• It is usually applied for gas transfer by bubble

formation in liquid.

• Therefore, we use the Lewis-Whitman (1923)

two-film model as described below.

Two film model

• It assumes that there are two stagnant layers

of film at the interface.

• The mass transfer across the films is by

molecular diffusion.

• The bulks of gas and liquid are homogeneous

with respect to solute.

Assuming that the mass transfer is in the X direction, in

other words from bulk gas phase to bulk liquid phase.

uniform

concentration

and pressure

because of

turbulence

uniform

concentration

because of

turbulence

• Since the solute is diffusing from the gas

phase into the liquid, there must be a

concentration gradient in the direction of

mass-transfer within each phase.

• The bulk gas concentration Cg falls to Cgi at

the interface.

• Similarly, in the liquid phase, the

concentration falls from Cli at the interface to

Cl in the main body of liquid.

• The bulk gas concentrations Cg and Cl are

clearly not the equilibrium values since

otherwise the mass transfer would not occur.

the interface.

Now, the flux from the bulk gas to the interface is

where

Jg = kg(Cgi-Cg)

Jl = kl(Cl -Cli)

and the flux from the interface to the bulk liquid is

klDl n

d

kg

Dg

nd

kl or kg = mass transfer coefficient for either the liquid

or gas phase as (dimensions = Length/time).

The same assumptions apply to the two films as apply

in the single Nernst film model.

So at steady state, the flux from the bulk gas to the

interface is equal to the the flux from the interface to

the bulk liquid.

Therefore,

kg(Cgi-Cg) = kl(Cl -Cli)

(C C )gi g

(C C )l

kl = kg li

Jg = Jl

• The difficulty is finding interface concentrations,

Cgi or Cli.

• We cannot directly measure the interface

concentrations.

• We want to be able to relate flux to bulk air and

water concentrations.

• Therefore, overall mass transfer coefficients

are defined.

• These coefficients are based on the

difference between the bulk concentration in

one phase and the concentration that would

be in equilibrium with the bulk concentration

in the other phase.

0

0,002

0,004

0,006

0,008

0,01

0,012

0,014

0,016

0 1 2 3 4 5 6

Cg

Cl

g i c l iC H C

Cl*

Cg*

Cli Cl

Cgi

Cg

Cg

Cl

Equilibrium line A

B

In the previous graph, point A represents the two

bulk phase concentrations and point B represents

the concentrations at the interface.

Define:

J

J

*K C Cl ll

*K C Cg g g

Kl = overall mass transfer coefficient based on

liquid-phase concentration.

Kg = overall mass transfer coefficient based on

gas-phase concentration.

Cli is replaced with Cl*

Cgi is replaced with Cg*

Kg,l have dimensions of Length/t.

Cl* = liquid phase concentration that would be in

equilibrium with the bulk gas

concentration.

= Cg/Hc (typical dimensions are moles/m3).

Cg* = gas phase concentration that would be in

equilibrium with the bulk liquid

concentration.

= HcCl (typical dimensions are moles/m3).

Expand the liquid-phase overall flux equation to

include the interface liquid concentration.

*J K C C C Cll l li li

Then substitute

CgiCli Hc

and Cg*ClHc

to get:

C Cggi

J K C Cl lil Hc

In the steady-state, fluxes through all films must

be equal. Let all these fluxes be equal to J.

C Cl liJ

kl

and

C Cgi gJ

kg

Jg = kg(Cgi-Cg) Jl = kl(Cl -Cli)

On an individual film basis:

Since all J’s are equal:

J JJ Kl k H kl c g

This can be arranged to give:

1 1

k gl kl

1

K Hc

Hc : Contaminant specific

kl and kg : Environment specific, such as mixing

A similar manipulation starting with the overall flux

equation based on gas phase concentration will give:

H 1c

k k

1

g l gK

These last two equations can be viewed as

"resistance" expressions where 1/Kg or 1/Kl

represent total resistance to mass transfer based

on gas or liquid phase concentration,

respectively.

In fact, the total resistance to transfer is made up of

three series resistances:

i) liquid film,

ii) interface

iii) gas film.

However, we assume instant equilibrium at the

interface so there is no transfer limitation here. It

should be noted that model selection (penetration,

surface renewal or film) does not influence the

outcome of this analysis.

The Two Special Cases - Single film control

• It is possible that one of the films exhibits

relatively high resistance and therefore

dominates the overall resistance to transfer.

• This, of course, depends on the relative

magnitudes of kl, kg and Hc.

• So the solubility of the gas and the

hydrodynamic conditions which establish the

film thickness determine if a film controls.

For highly soluble gases such as NH3 and SO2

(Hc is low)

1 1

k H kl c g

1

Kl g

1

H kc

Hcl gK k

Mass transfer is controlled by gas film resistance.

In general, slightly soluble gases such as N2 and

O2 (Hc is high)

1 1

k gl kl

1

K Hc

1 1

k H kl c g

1

Kl

1

kl or llK k

Mass transfer is controlled by liquid film resistance.