Modeling the Integration of Adsorption with Biological Processes in Wastewater Treatment - Activated Carbon for Water and Wastewater Treatment - page 166

Þ @ q
@

S b
@

2 S b
@

S b

z r p ð

1 e

@

Pe @

1

z 2 @

B

t

t ¼

ð EBCT Þ

@ S 0

(6.73)

3 ð 1 e Þ B

k max X f L f
S 0

R p

ð EBCT Þ

ð e B Þ

R p

3 1

k max X f L f
S 0

S b
@

2 S b

S b q @

Pe @

1

@ D g @

t ¼ z t

ð EBCT Þ (6.74) 2

@ @

@ z

The third term on the right-hand side of Eq. (6.74) stands for the dimensionless
adsorption and is composed of terms related to adsorption equilibrium and
adsorption kinetics. Similarly, the fourth term shows the dimensionless biofilm
removal. Depending on the stage of operation, either adsorption or biodegradation
or both will be of importance in the reactor. For example, when a biofilm has not
developed yet (L f =0), the last term will be equal to zero.
Alternatively, Eq. (6.38), which does not differentiate between adsorptive and
biological removal, may be written in dimensionless terms:

2

S b
@

2 S b

S b @

Pe @

1

@

L B
v i

3 ð 1 e Þ ð R p þ L f Þ B

S b S s Þ

t ¼ z

k fc ð (6.75)

@ z

2

@ e R p

B

The last term on the right-hand side shows the liquid film mass transfer to the
surface of biofilm. Assuming that the thickness of biofilm is negligibly small, this
equation takes the following form:

S b
@

2 S b

S b

@

Pe @

1

@

L B
v i

3 ð 1 e Þ

B

S b S s Þ

t ¼ z

k fc ð (6.76)

z 2

@ e

@

B R p

The last term on the right-hand side of this equation contains the Stanton
number, which represents the ratio between the liquid film transfer and the
advective (bulk) transport in the reactor [6, 15]:

3 ð 1 e Þ k fc L B B

Stanton number ð St Þ¼ (6.77)

e B v i R p

However, in modeling, often the modified Stanton number is used [15]:

St Þ¼ ð

1 e Þ k fc L B B

Modified Stanton number ð (6.78)

e B v i R p

Accordingly, Eq. (6.76) can be written as follows:

S b
@

2 S b

S b

@

Pe @

1

@ 3St ð S b S s Þ

t ¼ z

(6.79)

@ z

2

@

6.1.4.2.7 Importance of Dimensionless Numbers in a Biofilm-Carbon System As
shown in Table 6.1, a number of dimensionless groups can be used in the char-
acterization ofGAC reactorswith biological activity (BAC reactors). Complementary to