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Unformatted text preview: of the energy
grade line (EGL) above the channel bottom. For a given flow rate, there are usually
two states possible for the same specific energy. Rectangular Channels Consider the possible states at a given location. Let q Q/b Vy be the discharge per
unit width of a rectangular channel. Then, with q constant, Eq. (10.28) becomes
E q2
2gy2 y Q
b q (10.29) Figure 10.8b is a plot of y versus E for constant q from Eq. (10.29). There is a minimum value of E at a certain value of y called the critical depth. By setting dE/dy 0
at constant q, we find that Emin occurs at
y q2
g yc Q2
b2g 1/3 1/3 (10.30) The associated minimum energy is
Emin 3
2c E(yc) y (10.31) The depth yc corresponds to channel velocity equal to the shallowwater wave propagation speed C0 from Eq. (10.10). To see this, rewrite Eq. (10.30) as
q2 3
gyc (gyc)y2
c V2y2
cc (10.32) By comparison it follows that the critical channel velocity is
Vc (gyc)1/2 C0 Fr 1 (10.33) For E Emin no solution exists in Fig. 10.8b, and thus such a flow...
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 Spring '08
 Sakar

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