Gas Transfer for Water Treatment - Concept and Fundamental

DEFINITION AND TERMS
• Gas transfer  a physical phenomenon,
by which gas molecules are exchanged
between a liquid and a gas at a gas-liquid
interface 
(1) an increase of the concentration of the
gas(es) in the liquid phase as long as this
phase is not saturated with the gas under
the given conditions of e.g. pressure,
temperature (absorption of gas)
(2) a decrease when the liquid phase is
over saturated (desorption, precipitation
or stripping of gas)

DEFINITION AND TERMS
• Important natural phenomena of gas transfer 
the reaeration of surface water:
(1) the transfer of oxygen into surface water
(2) release of oxygen produced by algal activities
up to a concentration above the saturation
concentration
(3) release of taste and odor-producing
substances
(4) release of methane, hydrogen sulfide under
anaerobic conditions of surface water or of the
bottom deposits

ELEMENTS OF AERATION AND
GAS TRANSFER OPERATIONS
• Gas transfer occurs only through the
gas-liquid interface  has to be
carried out as to maximize the
opportunity of interfacial contact
between the two phases.
• The engineering goal  to
accomplish the gas transfer with a
minimum expenditure of initial and
operational cost (energy).

• Four different types of aerators:
(1) Gravity aerators
(a) cascades  the available difference
head is subdivided into several steps
(b) inclined planes  eqipped with riffle
plates to break up the sheet of water for
surface renewal
(c) vertical stacks  droplets fall and
updrafts of air ascend in counter current
flow

ELEMENTS –
AMMONIA STRIPPING

(2) Spray aerators
 the water is sprayed in the form
of fine droplets into the air 
creating a large gas-liquid interface
for gas transfer

(3) Air diffusers (bubble aeration)
 air is injected into water
(a) through orifices or nozzles in the air
piping system
(b) through spargers
(c) through porous tubes, plates, boxes or
domes
 to produce bubbles of various size
with different interfacial areas per m3 of
air.

ELEMENTS –
AIR DIFFUSERS (POROUS TUBES)

(4) Mechanical aerators
 create new gas-liquid interfaces
by different means and constructions
 two types of construction:
(a) various construction of brushes 
a horizontal revolving shaft with
combs, blades or angles
(b) turbine or cone aerators with
vertical shaft

Boyle’s Law
constant (constant temperature)
pV 

Charles’ Law
constant (constant pressure)
V
T


Gay-Lussac’s Law
constant (constant volume)
p
T


Ideal Gas Law
The ideal gas law is a special form of an equation of state,
i.e., an equation relating the variables that characterize a gas
(pressure, volume, temperature, density, ….).
The ideal gas law is applicable to low-density gases.
constant (fixed mass of gas)
B
pV
T
pV nRT
pV Nk T
p RT






Absolute Zero and the Kelvin
Scale
The pressure-temperature relation leads to the design of a
constant-volume gas thermometer.
Extrapolation of measurements made using different gases
leads to the concept of absolute zero, when the pressure
(or volume) is zero.

Kinetic Theory: Applications
Kinetic theory investigates (on a molecular scale) topics such as:
•Change of phase (evaporation; vapour pressure; latent heat)
•Pressure
•Change of shape and volume (elasticity; Hooke's law)
•Transport phenomena (diffusion - transport of mass; viscosity -
transport of momentum; electrical conduction - transport of electric
charge; thermal conduction - transport of heat)
•Thermal expansion
•Surface energy and surface tension

Kinetic Theory of Gases: Basic
Assumptions
•The number of molecules is large, and the average separation
between them is large compared with their dimensions. This means that
the molecules occupy a negligible volume in the container.
•The molecules obey Newton's laws of motion, but as a whole they
move randomly. 'Randomly' means that any molecule can move equally in any direction.
•The molecules undergo elastic collisions with each other and with
the walls of the container. Thus, in the collisions both kinetic energy and momentum are
constant.
•The forces between molecules are negligible except during a
collision. The forces between a molecule are short-range, so the molecules interact with each
other only during a collision.
•The gas is a pure substance. All molecules are identical.

SOLUBILITY OF GASES
• The solubility of gases in water (and also
in other liquids) depends upon:
(1) the nature of the gas generally
expressed by a gas specific coefficient 
the distribution coefficient, kD
(2) the concentration of the respective gas
in the gaseous phase  related to the
partial pressure of the respective gas in
the gas phase
(3) the temperature of the water
(4) impurities contained in the water

INFLUENCE OF THE GAS
CONCENTRATION ON SOLUBILITY
• The higher the gas concentration in
the gaseous phase  the greater will
be the saturation concentration in
the liquid phase
• The relation between the saturation
concentration cs (g/m3) and the gas
concentration in the gas phase cg
(g/m3):
cs = kD . cg

• The molar gas concentration in the gas
phase (according to the universal gas
law):
(n/V) = p / (RT) (moles/m3)
• Hence the corresponding mass
concentration cg is obtained by
multiplication with the molecular weight
(MW) of the gas:
cg = (p . MW)/ (RT) (g/m3)

• The combination yields:
cs = (kD . MW . p)/ (RT)
• Henry’s law is generally written as:
cs = kH . p
• The relation between distribution
coefficient kD and Henry’s constant:
kH = (kD . MW)/ (RT)

• Bunsen absorption coefficient, kb 
how much gas volume (m3), reduced
to standard temperature (0oC) and
pressure (101,3 kPa), can be
absorbed per unit volume (m3) of
water at a partial pressure of pO =
101,3 kPa of the gas in the gas
phase :
cs (m3
STP gas/m3 water) = kb

• And any other partial pressure p:
cs = kb . (p/p0) (m3
STP/m3)
• Since 1 m3
STP contains p0/R.T0 moles
of gas and a mass of gas equal to
MW. p0/R.T0 :
cs = (kb . MW)/(R.T0 ) p (g/m3)

• The relation between kD and kb:
kb = kD T0/T
• The interrelationship between the
three coefficients:
kD = kH .R.T/MW = kb .T/T0

• In the practice of aeration the gas
phase will always be saturated with
water vapor exerting a certain partial
pressure pw  the partial pressure p
of the other gases are reduced 
p’ = p . (P – pw)/P

INFLUENCE OF TEMPERATURE
ON SOLUBILITY
• Gases dissolved in water  accompanied
by liberation of heat H
• Le Chatelier principle  increase of
temperature results in a decrease of
solubility  van’t Hoff’s equation:
[d(ln kD)/dT] = H/(RT2)
where R = universal gas constant
T = absolute temperature K
H = change of heat content
accompanying by the absorp-
tion of 1 mole of gas (J/mole)

INFLUENCE OF TEMPERATURE
ON SOLUBILITY
• By integrating between the limits T1
and T2:
ln[(kD)2/(kD)1]= (H/R)(T2-T1)/(T1.T2)
• The product T1 .T2 does not change
significantly within the temperature
range encountered in gas transfer
operations:
(kD)2= (kD)1. econst (T2 – T1)

INFLUENCE OF IMPURITIES
ON SOLUBILITY
• Other constituent that may be contained
in water influence the solubility of gases 
expressed by an activity coefficient  :
cs = (kD/).cg
• For pure water  = 1
  generally increases as the
concentration of substances dissolved in
water rises  lowering the solubility

INFLUENCE OF IMPURITIES
ON SOLUBILITY
• The influence of concentration of impurities cimp
on the activity coefficient:
for non-electrolytes
log  = f . Cimp
for electrolytes
log  = f . I
where f = a constant depending on the
matter dissolved in water
I = ionic strength of electrolyte

DIFFUSION
• The phenomenon of diffusion  the
tendency any substance the spread
uniformly throughout the space
available to it  in environmental
engineering  diffusion phenomena
the liquid phase in gas transfer
operations

DIFFUSION
• For a quiescent body of water of unlimited
depth contacting the gas by an area of A
 the rate of mass transfer dM/dt as a
consequence of diffusion of the gas
molecules in the liquid phase  Fick’s Law
(dM/dt) = -D.A (dc/dx) (g/s)
where
D = coefficient of molecular diffusion (m2/s)
x = the distance from the interfacial area A
dx/dt = concentration gradient

DIFFUSION
• The total amount of gas M (g) that has
been absorbed through the surface area
A during the time t  independent of x

under conditions of unlimited depth of
water body

Dt
c
c
A
M s )
(
2 0



DIFFUSION
• If the depth is not too small  the time
of diffusion is not too long  diffusion is
very slow process and only very little gas
is brought into deeper layers of the water
body:
t
D
c
c
A
dt
dM
s 
)
( 0



THE CONCEPT OF GAS
TRANSFER COEFFICIENTS

THE CONCEPT OF GAS
• In accordance with Fick’s Law  the
mass transport per unit time (g/s) is
proportional to the concentration
difference :
for the gas phase
for the liquid phase
)
( gi
g
g c
c
A
k
m 

)
( L
Li
L c
c
A
k
m 


THE CONCEPT OF GAS
where
kg = partial gas transfer coefficient for
the gas phase
kL = partial gas transfer coefficient for
the liquid phase
cgi and cLi  generally not known 
cLi = kD . cgi

)
(
)
1
( 1
L
g
D
g
D
L
c
c
k
A
k
k
k
m 

 

THE CONCEPT OF GAS
• The total gas transfer coefficient KL is
composed of both the partial coefficients
and the distribution coefficient:
then
m = A KL (kDcg – cL)
g
D
L
L k
k
k
K


1
1

THE CONCEPT OF GAS
• The value of kD/kg will be very small
with respect to 1/kL  the influence
of the gas transfer coefficient of the
gas phase may be neglected 
KL = kL
and consequently
m = A kL (kDcg – cL)

PENETRATION THEORY
• During the time of exposure the gas
diffuses into the fluid element 
penetrates into liquid.
• In contrast to the film theory, the
penetration process is described by
unsteady diffusion

PENETRATION THEORY
• During the time of the liquid the interface
to the gas, the gases penetrate into the
liquid at a diminishing rate. The total
mass of gas absorbed during this time:

Dt
c
c
k
A
M L
g
D )
(
2 


PENETRATION THEORY
• Hence the average absorption rate m
(g/s) during the time t is defined by
The penetration assumes
t =tc
for a gas transfer process operated under
steady state condition
t
D
c
c
k
A
m
t
M
L
g
D

)
(
2 



PENETRATION THEORY
• The final form of the rate expression for
gas absorption as proposed by the
penetration theory:
)
(
2 L
g
D
c
c
c
k
A
t
D
m 



PENETRATION THEORY
• According to the penetration theory:
stating that the coefficient of gas
transfer is proportional to the root of
the coefficient diffusion.
c
L
t
D
k

2


PENETRATION THEORY
• Assumption of a constant time of exposure of
fluid elements to the gas phase  a constant
rate rc (s-1)
• Taking rc instead of tc
c
c t
r 1


c
L
Dr
k 2


SURFACE RENEWAL THEORY
• The model underlying the surface renewal
theory is equal to that of the penetration
theory  unsteady diffusion of the gas
into liquid elements exposed to the gas
phase.
• However, this theory does not assume that
the time to be constant  follow a
frequency distribution f(t) with ages of the
fluid elements (= time of exposure)
ranging from zero to infinity.

• The theory is based on the assumption  the
fraction of the surface having ages between t
and t+dt is given by:
 if the surface element of any age always has
chance of s.dt of being replaced  if each
surface element is being renewed with a
frequency s, independent of its age
dt
se
dt
t
f st


)
(

• The average rate of gas transfer is
• The surface renewal theory forecasts
Ds
kL 

FILM-SURFACE-RENEWAL
THEORY
• This theory attempts a combination of the
film theory and the surface renewal
theory in principle  a combination of
steady and unsteady diffusion.
• The gas transfer coefficient as a function
of the rate of surface renewal s and max
x = dL
D
d
s
Ds
k L
L
2
.
coth


FACTORS AFFECTING THE GAS
• The effects of temperature on the rate
gas transfer (effects on kL and A)
• The temperature coefficient  for
oxygenation of sewage  in the range of
1,016 to 1,047.
1
2
1
2
.
)
(
)
( T
T
T
L
T
L
V
A
k
V
A
k 
 

• The influence of hydrophobic constituents and surface active
agents on the rate of gas transfer  Gibbs adsorption
equation
c = concentration of hydrophobic substance in the bulk of
the solution (g/m3)
S = excess concentration of hydrophobic substance at the
surface (g/m3) as compared with that of the bulk
solution
R = universal gas constant
d/dc = rate of increase of surface tension with increasing
the concentration of the hydrophobic substance
dc
d
RT
c
S




THE OVERALL GAS TRANSFER
COEFFICIENT OR AERATION
COEFFICIENT
• Under steady state conditions of gas
transfer operation  the coefficient
diffusion and the time of exposure may be
assumed constant :
where k2 or kL.a is the overall gas transfer
coefficient.
L
L
c
k
a
k
V
A
t
D
V
A
k .
2
2 




COEFFICIENT
• The rate of gas transfer can be
expressed as the rate of concentration
change
which integrates with c0 at t=0 to
or
 
c
c
k
dt
dc
V
m
s 

 2
  t
k
s
s e
c
c
c
c 2
0




t
k
s
s
e
c
c
c
c 2
0





COEFFICIENT
• The overall gas transfer coefficient k2
can easily determined experimentally by
measuring the change of concentration
as a function of time and by plotting log
(cs-c)/(cs-c0) versus time :
e
t
k
e
c
c
c
c t
k
s
s
log
.
log
log 2
0
2




 
t
k2
.
4343
,
0



THE EFFICIENCY
COEFFICIENT
• With some transfer operations, e.g. cascades,
weir aeration  difficult or impossible to
determine the parameter time t.
• If now a constant time tk is assumed for the
aeration step under steady state conditions:
K
e
c
c
c
c k
t
k
s
e
s




 
1
2
0
K
c
c
c
c
c
c
c
c
s
e
s
e
s








1
1
0
0
0

Gas Transfer for Water Treatment - Concept and Fundamental

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