The energy equation for the streamtube becomes

2

v 2 p2

v 1 p1

+ z1 = 2 +

+

+ z 2 + hl

(2.41)

2g γ

γ

2g

If there is no shear stress on the streamtube boundary and if there is no change in internal

energy (u1 = u2), the energy equation reduces to:

2

v 2 p2

v 1 p1

+

+ z1 = 2 +

+ z 2 = cons tan t

(2.42)

2g γ

2g γ

which is the Bernoulli Equation.

Generally, there is not sufficient information available to do a differential streamtube analysis of a

reach of river, so appropriate changes must be made in the energy equation. A reach of river

such as that shown in Figure 2.1 can be pictured as a bundle of streamtubes. We know the

statement of the conservation of energy for a streamtube. It is Equation 2.41 which can be

written:

v 1 p1

v 2 p2

2

+ z1 v dA = 2 +

+ z 2 v dA + hl vdA

+

(2.43)

2g γ

2g γ

because v1 dA1 = v2 dA2 = vdA for the streamtube.

The common form of the energy equation used in open channel flow is derived by integrating

Equation 2.43 over the cross-section area:

αV 2 p

v2 p

+ + z vdA =

+ + z Q

(2.44)

A 2g γ

γ

2g

where:

=

Kinetic energy correction factor defined by the expression

α

1

α=

v 3 dA

(2.45)

A

V3A

to allow the use of average velocity V rather than point velocity v. The average pressure over the

cross-section is p , defined as:

1

p=

p vdA

(2.46)

A

VA

2.11

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