12-1prepared by Ercan Kahya

Lecture Notes 3: (Chapter 12) Lecture Notes 3: (Chapter 12)

Energy principles in Open-Energy principles in Open-Channel Channel

Energy generated at an overfall (Niagara Falls). Energy generated at an overfall (Niagara Falls).

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Total & Specific Energy

Specific Energy: the energy per unit weight of water measured from the channel bottom as a datum

► Note that specific energy & total energy are not generally equally.

At section 1: Specific Energy Total Energy

12-3

Total & Specific Energy► Specific energy varies abruptly as does the channel geometry

► Velocity coefficient (α) is used to account nonuniformity of the velocity distribution when using average velocity.

► It varies from 1.05 (for uniform cross-sections) to 1.2 (nonuniform sections).

► For natural channels, a common method to estimate α:

A channel section divided into three sectionsA channel section divided into three sections

Weighted mean velocity:

12-4

Specific EnergyAssuming α equal to 1, it is convenient to express E in terms of Q for steady flow conditions

Specific Energy Diagram (SED)

f(E, Q, y) = 0

SED is a graphical representation for the variation of E with y.

Let`s write E equation in terms of static & kinetic energy:

where and

12-5

Specific Energy Diagram

The specific energy diagramThe specific energy diagram

- Es varies linearly with y

- Ek varies nonlinearly with y

- Horizontal sum of the line OD & the curve kk` produces SED

- For given E: alternate depths (y1 & y2)

- They are two depths with the same specific energy and conveying the same discharge

-Emin vs critical depth

12-6

Specific Energy Diagram

The specific energy diagramThe specific energy diagramfor various dischargesfor various discharges

- An increase in the required Emin yields bigger discharges.

- Fn : Froude number

equals to V square / gD

12-7

Critical Flow ConditionsGeneral mathematical formulation for critical flow conditions:General mathematical formulation for critical flow conditions:

- Assume dA/dy = B

12-8

Critical Flow ConditionsAt the critical flow conditions, specific energy is minimum:At the critical flow conditions, specific energy is minimum:

Then, which can also be expressed as -->

Then,

In wide or rectangular section, D = y

at critical depth

12-9

Critical Velocity

The general expressions forThe general expressions for

Used to determine the Used to determine the state of flow state of flow

Critical state condition:

Critical velocity for the general cross section:

Velocity head at critical conditions:

In wide or rectangular section, D = y

12-10

Critical DepthFor a certain section & given discharge:For a certain section & given discharge:

Critical depth Critical depth is defined as the depth of flow requiring minimum specific energy

This equation should be solved …

For the trapezoidal cross section:

Solve this by trial & error …

Critical depth Critical depth trapezoidal and circular trapezoidal and circular sections sections

For the rectangular cross section:

12-11

Critical Energy

Recall for any cross section:

Then,

For wide or rectangular section, D = y

Critical Energy Critical Energy is the energy when the flow is under critical conditions.

12-12

Critical Slope

For Chezy equation:

Then,

For direct computation:

Critical slope Critical slope is the bed slope of the channel producing critical conditions.

► depends discharge; channel geometry; resistance or roughness

For Manning equation: In English unit:

12-13

Critical Slope

Critical slope Critical slope is very important in open-channel hydraulics. WHY?

The The summarysummary given above encompasses much of the important concepts given above encompasses much of the important concepts

of the energy & resistance principles as applied to open channels.of the energy & resistance principles as applied to open channels.

3/42

22

RA

nQSc

12-14

Discharge-Depth Relation for Constant Specific Energy

Now assume Eo constant, then evaluate Q-y relation:

For the condition of the Qmax:

It reduces to

Then substitute this into Q equation at the top:

implies that the Qmax is encountered at the critical flow condition for given E.

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Discharge-Depth Relation for Constant Specific Energy

can be written as

Differentiating this w.r.t. y and equating to zero:

For wide or rectangular section, D = y

Q-yQ-y relation relation

for constant specific energyfor constant specific energy

12-16

Transitions in Channel BedsConsider Consider an open-channel with a small drop ∆z in its bed

A small drop in the channel bed (subcritical flow): (a) change in A small drop in the channel bed (subcritical flow): (a) change in water levels, and (b) steps for solution.water levels, and (b) steps for solution.

Assume that friction losses and minor losses due to drop are negligible

The method provides a good first approximation of the effects of the transition

First step: First step: compare the given conditions to critical conditions to determine the initial state of flow.

12-17

Transitions in Channel Beds

Consider Consider an abrupt rise ∆z in the open-channel bed

Assume that upstream conditions are subcritical & initial E1

Note that ∆z should be subtracted from E1 & While TEL unchanged, E reduced

12-18

Transitions in Channel Beds

Consider Consider an abrupt rise ∆z in the open-channel bed

Assume that upstream conditions are supercritical & initial E1

Note that ∆z should be subtracted from E1 & While TEL unchanged, E reduced

RESULT : Water depth must rise after the step

ChokesChokes

Chokes can only occur Chokes can only occur when the when the channel is constrictedchannel is constricted, but will not , but will not occur where the flow area expanded occur where the flow area expanded such as drops or expansions. such as drops or expansions.

In designing a channel transition In designing a channel transition that would tend to restrict the flow, that would tend to restrict the flow, engineer wants to engineer wants to avoid forcing a avoid forcing a choke choke to occur if at all possible. to occur if at all possible.

ChokesChokes

Figure 12.16: Rise in a channel bed: (a) a small step-up, (b) a bigger step-up

ChokesChokes

Figure 12.16: Rise in a channel bed: (c) a still bigger step-up, and (d) changes in the specific energy.

(a) A contracted channel. (b) Water levels in a contracted channel. (c) SED for a contracted channel. (d) Water level in a contracted channel-supercritical flow.

(a)

(b)

(c)

(d)

Enlargements and constructions in channel widths

EXAMPLE

A 6.0 m rectangular channel carries a discharge of 30 m3/s at a depth of 2.5m. Determine the constricted channel width that produces critical depth.

EXAMPLE: s o l u t i o n

mgy

qyE 70.2

5.2*81.9*2

55.2

2 2

2

2

2

myEE c 80.1y 2

3cmin

smgyqg

qy cc / 56.7 23

3

2

b2 = Q/ q2 = 30 / 7.56 = 3.07 m

Weirs & Spillways

g

Vy

g

Vy

22

22

2

21

1

y2=0

2112 2 VgyV

Hg

Vy

2

21

1

gHV 22

To control the elevation of the water

- Functions as a downstream choke control

- Classified as sharp crested or broad cresteddepending on critical depth occurrence on the crest

Head on the weir crest

Orifice equation:

LdHgHVdAdQ 2 2/32/1 23

22 LHgdHHLgQ

2/32/323

2CLHLHgCQ d

Weirs & SpillwaysImmediate region of weir crestAssume V1=0

Discharge through the element: Integrate across the head (0 - H):

Total discharge across the weir:

Coefficient of Discharge

2/32/323

2CLHLHgCQ d

Losses due to the advent of the drawdown of the flow immediately upstream of the weir as well as any other friction or contraction losses;

To account for these losses, a coefficient of discharge Cd is introduced.

ZHCd /08.0611.0 (Henderson, 1966)

where, H is the head on the weir crest, Z is the height of the weir.

Use this equation up to H/Z = 2

Discharge Measurements• Weirs• Flume• Orifices

• Weirs and flumes not only require a simple head reading to measure

discharge but they can also pass large flow without causing the

upstream level to rise significantly and causing flooding.

Discharge Control

- Orifices are rather cumbersome for discharge measurements, but

they are very useful for discharge control

Practical Hydraulics by Melvyn KayCopyright © 1998 by E & FN Spon . All rights reserved.

Discharge Control

Practical Hydraulics by Melvyn KayCopyright © 1998 by E & FN Spon . All rights reserved.

WEIRS

WEIRS

FLUMES

Practical Hydraulics by Melvyn KayCopyright © 1998 by E & FN Spon . All rights reserved.

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Class Exercises: