Environmental Engineering Reference
In-Depth Information
Typical values of
k
a
at 20°C are given in Table 4.5, which
indicates that reaeration coefficients typically vary from
0.1 d
−1
for small ponds to greater than 1.15 d
−1
for rapids
and waterfalls. In small rivers, rapids play a major role
in maintaining high dissolved oxygen levels and elimi-
nating rapids by dredging or damming a river can have
a severe effect on Do. Several of the most popular
empirical formulas for estimating
k
a
at 20°C are given
in Table 4.6, where the units of
k
a
are d
−1
, the average
stream velocity,
V
, is in m/s
,
the flow rate,
Q
, is in m
3
/s,
the average stream depth,
d
and
D
, are in meters, the
stream width,
W
, is in meters, and the average channel
slope,
S
0
, and water-surface slope,
S
, are dimensionless.
For temperatures other than 20°C, use
similar to those of the o'Connor and Dobbins model,
but for faster streams. owens et al.'s (1964) formula is
used for shallower streams; in small streams, the formula
proposed by Tsivoglou and Wallace (1972) compares
best with observed values (Thomann and Mueller,
1987). The last four empirical formulas in Table 4.6 are
the most recent and are based on the most comprehen-
sive data set. According to their authors, they are the
most accurate, with standard errors in the range of 44-
61%, compared with standard errors of 65-115% in the
other methods (Bennett and Rathbun, 1972). The for-
mulas listed in Table 4.6 all give values of
k
a
that
approach zero as the depth of the stream increases,
implying that reaeration becomes negligible for deep
bodies of water. This is certainly not the case, since when
water motion is less significant, wind becomes the domi-
nating factor in reaeration. The reaeration constant
typically has a minimum value in the range
k
=
k
θ
T
−
20
(4.60)
a
a
T
20
where
T
is the temperature of the stream (°C),
k
a
T
and
k
a
20
are the values of
k
a
at temperatures
T
and 20°C
respectively, and
θ
is the temperature coefficient, which
is commonly taken to be in the range of 1.024-1.025. In
practice, the empirical formula proposed by o'Connor
and Dobbins, (1958) has the widest applicability and
provides reasonable estimates of
k
a
at 20°C in most
cases. Churchill et al.'s (1962) formula applies to depths
=
0.6
1.0
k
to
(4.61)
a
min
d
d
If the calculated value of
k
a
falls below the range
of
minimum values given in Equation (4.61),
k
a
= 0.6/
d
should be used.
TABLE 4.5. Typical Reaeration Constants
EXAMPLE 4.10
Ranges of
k
a
at 20°C (d
−1
)
Water Body
A river with riffles and pools has a width of 20 m, a
mean depth of 5 m, a slope of 0.00003, and an estimated
flow rate of 47 m
3
/s. (a) Estimate the reaeration
constant using the applicable equation(s) in Table 4.6.
(b) If the temperature of the river is 20°C and the dis-
solved oxygen concentration is 5 mg/L, estimate the
reaeration rate. (c) Determine the mass of oxygen added
per day per meter along the river.
Small Ponds and backwaters
0.10-0.23
Sluggish streams and large lakes
0.23-0.35
Large streams of low velocity
0.35-0.46
Large streams of normal velocity
0.46-0.69
Swift streams
0.69-1.15
Rapids and waterfalls
>1.15
Sources of data
: Tchobanoglous and Schroeder (1985).
TABLE 4.6. Empirical Formulas for Estimating Reaeration Constant,
k
a
at 20°C
Formula
Field Conditions
Reference
0.5
V
d
k
a
= 3.93
o'Connor and Dobbins (1958)
0.3
m
<
d
<
9
m
, 0.15 m/s <
V
< 0.50 m/s
1.5
V
d
k
a
= 5.23
0.6
m
<
d
<
3
m
, 0.55 m/s <
V
< 1.50 m/s
Churchill et al. (1962)
1.67
V
d
0.67
1.85
k
a
= 5.32
0.1
m
<
d
<
3
m
, 0.03 m/s <
V
< 1.50 m/s
owens et al. (1964)
k
a
= 3.1 × 10
4
VS
0
0.3
m
<
d
<
0.9
m
, 0.03 m
3
/s <
Q
< 0.3 m
3
/s
Tsivoglou and Wallace (1972)
k
a
= 517(
VS
)
0.524
Q
−0.242
Pool and riffle streams,
Q
< 0.556 m
3
/s
Melching and Flores (1999)
k
a
= 596(
VS
)
0.528
Q
−0.136
Pool and riffle streams,
Q
> 0.556 m
3
/s
Melching and Flores (1999)
k
a
= 88(
VS
)
0.313
D
−0.353
Channel-control streams,
Q
< 0.556 m
3
/s
Melching and Flores (1999)
k
a
= 142(
VS
)
0.333
D
−0.66
W
−0.243
Channel-control streams,
Q
> 0.556 m
3
/s
Melching and Flores (1999)
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