Applications of Chemical Kinetics in Environmental Systems - Elements of Environmental Engineering: Thermodynamics and Kinetics - page 301

is τ = V a /Q g . Hence upon integration, we obtain the following:

ln C w
C w

Q g
V w

K aw 1

e − ((K w a)(V w /Q g )( 1 /K aw )) t =− − =− k rem t , (6.99)

where k rem is the first-order removal rate constant from the aqueous phase. One can
define partial gas-phase saturation as φ =

1 − exp ( −φ H s ) , where

(K w a) V w
Q g

1
K aw H s

.

φ =

In this definition, H s is the depth at which the air bubble is released in the lagoon.
Thus we can rewrite the equation for k rem as follows:

Q g
V w

K aw ( 1 −

e −φ H s ) . k rem = (6.100)

Two special limiting cases are to be noted:

(a) If the exit gas is saturated and is in equilibrium with the aqueous phase,
1 − exp ( −φ H s ) → 1 and k rem = (Q g /V w )K aw . This condition can be satis-

fied if ( K w a ) is large.
(b) The second limiting case is (K w a)(V w /Q g )( 1 /K aw ) 1, for which k rem =

(K w a) . This represents the case when the exit gas is far from saturation. For
large K aw and Q g values, this limiting case will apply. This is the case in most
surface aeration systems, where a large volumetric flow of air is in contact
with an aqueous body.

Let us now consider a continuous flow system. If the continuous flow system is
in plug flow , then at steady state the ideal residence time for the aqueous phase is
t V w /Q L ,andsubstituting C w =

C 0 intheaboveequationforabatchreactorshould
give us the appropriate equation for a PFR bubble column:
ln C w
C 0

=

Q g
Q L

K aw ( 1

e −φ H s ) . =− − (6.101)

The term ( Q g /Q L )K aw = S is called the separation factor in chemical engineering,
and gives the maximum separation achievable if the exit air is in equilibrium with the
aqueous phase.
If the mixing in the aqueous phase is large, a CSTR approach can be used to model
the process. The overall mass balance is then given by

C g −

) .

d C w
d t =

Q L
V w

Q g
V w

(C 0 − C w ) + C g ( τ (6.102)

Since the entering gas is always clean, C g =

0. Assuming steady state, d C w / d t = 0.

Therefore, we obtain

C w
C 0 =

1
1 + S( 1 −

e −φ H s ) .

(6.103)