1282 WATER FLOW
Thus, the energy per unit weight of water passing the cross-
section per second, H is:HZDV
22 g. (47)
The term, z D V^2 2 g is known as the total head or total
energy level (TEL); the latter name is used here. The slope
of the total energy level line is the energy gradient or friction
slope and gives the rate of energy dissipation in the flow.
The energies at Sections 1 and 2 are related by the
expression:zDV
ZDV QN(^11) AR
1
2
22
2
2 22
22 gg 24 / (48)
in which the Manning equation is used to calculate the fric-
tion slope and the mean values for the flow area, A, and the
hydraulic radius, R, are to be used.
Flow Regimes
Critical Flow The specific energy, E, is defined as the total
head referred to the channel bottom (Figure 6):
ED
Q
A
1
2 g^2. (49)
Differentiating Eq. (47) with respect to D and equating the
derivative to zero to obtain its minimum value,
d
d
d
d
E
D
Q
A
A
D
10
2
g^3. (50)
Noting that d A T d D, Eq. (50) becomes:
QA
T
23
g
. (51)
Equation (51) is the fundamental equation for critical flow
and is applicable to all shapes of cross-sections.
If the mean depth of the flow section is defined as
Dm A/T, substitution of this relation into Eq. (49) would
give the significant expressions:
VDcm^2
22 g
(52)
and
V
D
c
g m
1. (53)^
At critical flow, Eq. (52) demonstrates that the velocity head
equals one-half the mean depth and Eq. (53) indicates that
the Froude number equals unity.
Specific Energy Diagram for Rectangular Channel For a
rectangular channel, Q qB in which q discharge per unit
width, B channel width, and Eq. (49) becomes,
ED
q
D
2
2 g^2. (54)
A plot of Eq. (54) for any given constant unit discharge gives
Figure 7, which is known as the specific energy diagram. The
TEL
E D
T
D dD dA
V^2 /2g
FIGURE 6 Derivation of critical flow.
45° line
E = D
Flow depth, D
Critical line q 1
q 2
Supercritical
Ec = D^32 c
Subcritical
FIGURE 7 Specific energy diagram.
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