Applications of Chemical Kinetics in Environmental Systems - Elements of Environmental Engineering: Thermodynamics and Kinetics - page 278

2

Ideal plug flow

1.5

D ax / uL

1

0.02

0.2

0.5

0.5

0

0 0.5 1 1.5 2 2.5

t / t

FIGURE 6.7 Relationship between F(t/ τ ) and dimensionless time for different values of
D ax /uL . True plug-flow model is represented by the vertical line at dimensionless time of 1.0.

The term D ax /uL is the dispersion number .
If a perfect pulse is introduced in a flowing fluid, the solution to the above equation
gives the exit concentration (Hill, 1977)

C(L , t)
0 C(L , t) d (t/

1

) e − (( 1 − (t/ τ )) 2 /( 4 D ax /uL) · (t/ τ )) . ) = 2 V R √ (

(6.28)

π τ

τ D ax /uL)(t/

τ

is given in Figure 6.7.
For small values of D ax /uL , the curve approaches that of a normal Gaussian error
curve. However, with increasing D ax /uL ( > 0.01), the shape changes significantly
over time. For small values of dispersion ( D ax /uL < 0.01), it is possible to calculate
D ax by plotting log C(L , t)/C( 0 ) versus t and obtaining the standard deviation

A plot of the right-hand side of the above equation versus t/

σ from

the data. C (0) is the influent pulse input concentration. The equation is

2 u 3

1
2 σ

D ax =

L .

(6.29)

6.1.2.2 Tanks-in-Series Model

The second model for dispersion is a series of CSTRs in series. The actual reactor is
then composed of n CSTRs; the total volume is V R = nV CSTR . The average residence

time in the actual reactor is nV CSTR /Q 0 , where Q 0 is the volumet-

ric flow rate into the reactor. For any reactor n in the series, the following material
balance holds:

τ = V R /Q 0 =

d C n
d t .

C n − 1 Q 0 = C n Q 0 + V CSTR (6.30)