WETLANDS - Water Quality Engineering in Natural Systems - page 299

depth (L), and N is the number of tanks (dimension-
less). In applying Equation (8.21) to real wetlands, the
parameter values are treated as empirical fitting param-
eters (Kadlec, 2003), and Equation (8.21) is commonly
expressed as

0 385 .

3 2 /

Q
C L H H

∆

= 1 −

( − ) /

3 2

H H −

w 0 w 0 w

0 38 . 5

3 2 /

1 7 . 0 05

0 30

.

= 1 −

( . 1 83 30 )( )( H − 0 30 . )

3 2 /

H − .

− P

*

0 0

C C
C C

−
−

k
Pq

= 1 + (8.22)

*

i

which yields H 0 = 0.41 m. Since ( H 0 − H w )/ H w =
0.37 ≤ 0.4, the assumption that C w = 1.83 is justified, and
the depth just upstream of the weir is 0.41 m. Hence, as
the downstream water surface rises from below the crest
of the weir to 5 cm above the crest of the weir and the
flow rate over the weir remains constant, the water
surface just upstream of the weir rises by 0.41 m −
0.40 m = 0.01 m = 1 cm.

where q is the design overflow rate ( = y /τ ), and P is
the apparent number of tanks in series (TIS). In apply-
ing Equation (8.22) to observed data, the three
adjustable parameters are P , k , and C *, and hence Equa-
tion (8.22) is sometimes called the P - k - C * model
(Kadlec and Wallace, 2009). The value of the areal rate
coefficient, k , is generally temperature dependent and
is described by the Arrhenius equation

(8.23)

k = k θ

T − 20

8.3.2.2 Performance-Based Sizing. The preferred
model for describing wetland hydraulics and contami-
nant attenuation in constructed wetlands is the tanks-
in-series (TIS) model (Kadlec and Wallace, 2009)
illustrated in Figure 8.14. The TIS model conceptualizes
a constructed wetland as a series of continuously stirred
tank reactors. For the case of steady flow with no water
losses or gains, the mass balance for the j th tank is
given by

20

where k 20 is the value of k at 20°C, and θ is a factor that
depends on the constituent of interest. Equation (8.22)
contains the Damköhler number, Da, defined in this
context as

Da = k

q

(8.24)

where Da measures the ratio of the advection time scale
relative to the decay time scale, and decay is significant
when Da > 1, which is almost always the case in wet-
lands (this is really why we use them).
In cases where there are large differences between
the wetland inflow and outflow ( Q o / Q i > 2 or Q o / Q i < 0.5),
the use of an average flow rate, Q , in Equation (8.22)
might not be appropriate. In these cases, the water
balance for the j th segment is given by

*

QC − QC = kA C C ( − ) (8.20)

j − 1 j j

where Q is the flow rate through the system (L3T−1), 3 T −1 ), C j −1
is the inflow concentration (ML −3 ), C j is the outflow
concentration (ML −3 ), k is the areal rate coefficient
(LT −1 ), A is the area (L 2 ), and C * is the background
concentration (ML −3 ). For a sequence of tanks the mass
balances given by Equation (8.20) combine to give

− N

Q Q = + A R ( − ET − I )

C C
C C

−
−

*

k
Ny

τ (8.25) j j −1 j

(8.21)

= 1 +

*

i

where Q j is the outflow (L3T−1), 3 T −1 ), Q j −1 is the inflow (L3T−1), 3 T −1 ),
A j is the area (L 2 ), R is the rainfall (L), ET is the
evapotranspiration (L), and I is the infiltration (L).
In determining the mass balance of a particular

where C is the concentration of the flow exiting the
system (ML −3 ), C i is the concentr a tion of the inflow
(ML −3 ), τ is the detention time (T), y is the average flow

P ET P ET P ET P ET

Q 1 , C 1 Q 2 , C 2 Q 3 , C 3

Tank 1 Tank 2 Tank 3 Tank N Q N , C N

Q in , C in

kA(C 1 - C*) kA(C 2 - C*) kA(C 3 - C*) kA(C N - C*)

Figure 8.14. Wetland mixing model.