Geoscience Reference
In-Depth Information
Inserting Eq. (3.94) into Eq. (3.98) leads to
q
b
∗
k
2.303 log
30.2
h
I
2
k
q
s
∗
k
=
∗
I
1
k
+
(3.99)
s
The fractional transport rate of bed-material load is then obtained by
q
t
∗
k
=
q
b
∗
k
+
q
s
∗
k
.
Einstein's method is an important contribution to sediment research. However, it is
laborious because numerical integrations are involved and
U
∗
is determined using his
movable bed roughness formula that needs to be solved iteratively. Many tests have
shown that Einstein's method can provide reasonable results for narrowly graded
sediment mixtures, but not for those widely graded (Misri
et al
., 1984; Samaga
et al
.,
1986a&b). Modifications were proposed by several investigators, such as Colby and
Hembree (1955), Toffaletti (1968), and Shen and Hung (1983). Van Rijn (1984b)
also established a similar method to calculate the suspended-load transport rate using
Eq. (3.97) with his distribution function (3.90) and near-bed concentration formula
(3.95).
Bagnold formula
Based on his stream power concept, Bagnold (1966) established the following formula
to calculate the suspended-load transport rate:
τ
b
U
2
ω
0.01
ρ
s
ρ
q
s
∗
=
(3.100)
−
ρ
s
s
where
q
s
∗
is the suspended-load transport rate by weight per unit time and width
m
−
1
s
−
1
).
(N
·
Zhang formula
Based on the energy balance of sediment-laden flow, Zhang (1961; also see Zhang
and Xie, 1993) derived the relation between suspended-load transport capacity
C
∗
and
parameter
U
3
/(
ω
s
)
, as shown in Fig. 3.21, using measured data from the Yangtze
River, the Yellow River, etc. Here,
C
gR
is the average suspended-load concentration
∗
m
−
3
). One may write the Zhang formula as Eq. (2.140) with variable coefficients
K
∗
and
m
. For convenience, Guo (2002) approximated the
C
∗
∼
(kg
·
U
3
/(
gR
ω
s
)
curve in
Fig. 3.21 by the following equation:
1.5
⎡
⎣
1
1.15
⎤
⎦
U
3
gR
1
45
U
3
gR
1
20
C
∗
=
+
(3.101)
ω
ω
s
s
Wu and Li (1992) extended the Zhang formula to determine the fractional concen-
tration of non-uniform suspended load as
C
p
bk
C
k
. Here,
p
bk
is the bed-material
gradation, and
C
k
is the potential equilibrium concentration of size class
k
determined
=
∗
k
