1
β′ =
y
2
(2.105)
y' v dy
o
V (y o - y′)
2
If we substitute Equations 2.78 and 2.104 into Equation 2.107 and integrate, the result is the
y 2
2y′
yo
yo
1
β′ =
o
In - 2 In + 2 -
(2.106)
y′
2
y o - y′
y′
yo
y
In 11.11 o
ks
Similarly, the energy coefficient for a vertical section unit width is
1
α′ =
yo
v 3 dy
(2.107)
y′
V 3 (y o - y′)
or
y 3
6y′
2
yo
yo
yo
1
α′ =
In - 3
In
+ 6 In - 6 +
o
(2.108)
y′
y′
y o - y′
2
y′
yo
y
In 11.11 o
ks
These equations (Equations 2.106 and 2.108 are rather complex, so a graph of α′ and β′ vs yo/ks
has been prepared. The relations are shown in Figure 2.12.
For the entire river cross-section (shown in Figure 2.13) Equation 2.45 can be written
[
0 v dy dw ]
1
y
W
3
α=
(2.109)
o
0
3
Q
A
A
where W is the top width of the section, w is the lateral location of any vertical section, yo is the
depth of flow at location w, and v is the local velocity at the position y, w. The total discharge is Q
and the total cross-sectional area is A.
2.29
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