Transcript Document

Lecture Notes 3: (Chapter 12)
Energy principles in Open-Channel
Energy generated at an overfall (Niagara Falls).
prepared by Ercan Kahya
12-0
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
12-1
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:
Weighted mean velocity:
A channel section divided into three sections
12-2
Specific Energy
Assuming α equal to 1, it is convenient to express E in terms of Q
for steady flow conditions
f(E, Q, y) = 0
Specific Energy Diagram (SED)
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
12-3
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
The specific energy diagram
12-4
Specific Energy Diagram
- An increase in the required
Emin yields bigger discharges.
- Fn : Froude number
equals to V square / gD
The specific energy diagram
for various discharges
12-5
Critical Flow Conditions
General mathematical formulation for critical flow conditions:
- Assume dA/dy = B
12-6
Critical Flow Conditions
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
12-7
Critical Velocity
The general expressions for
Used to determine the 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
12-8
Critical Depth
For a certain section & given discharge:
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 …
For the rectangular
cross section:
Critical depth trapezoidal and circular
sections
12-9
Critical Energy
Critical Energy is the energy when the flow is under critical conditions.
Recall for any cross section:
Then,
For wide or rectangular section, D = y
12-10
Critical Slope
Critical slope is the bed slope of the channel producing critical conditions.
► depends discharge; channel geometry; resistance or roughness
For Chezy equation:
Then,
For Manning equation:
In English unit:
For direct computation:
12-11
Critical Slope
Critical slope is very important in open-channel hydraulics. WHY?
2
2
Q n
Sc  2 4 / 3
A R
The summary given above encompasses much of the important concepts
of the energy & resistance principles as applied to open channels.
12-12
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.
12-13
Discharge-Depth Relation for Constant
Specific Energy
Q-y relation
for constant specific energy
For wide or rectangular section, D = y
can be written as
Differentiating this w.r.t. y and equating to zero:
12-14
Transitions in Channel Beds
Consider an open-channel with a small drop ∆z in its bed
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: compare the given conditions to critical conditions to determine
the initial state of flow.
A small drop in the channel bed (subcritical flow): (a) change in
water levels, and (b) steps for solution.
12-15
Transitions in Channel Beds
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
12-16
Transitions in Channel Beds
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
12-17
Chokes

Chokes can only occur when the channel is
constricted, but will not occur where the flow
area expanded such as drops or expansions.

In designing a channel transition that would
tend to restrict the flow, engineer wants to
avoid forcing a choke to occur if at all
possible.
Chokes
Figure 12.16: Rise in a channel bed: (a) a small step-up,
(b) a bigger step-up
Chokes
Figure 12.16: Rise in a channel bed: (c) a still bigger stepup, and (d) changes in the specific energy.
Enlargements and constructions in channel widths
(a)
(b)
(c)
(d)
(a) A contracted channel. (b) Water levels in a contracted channel.
(c) SED for a contracted channel. (d) Water level in a contracted channelsupercritical flow.
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:
solution
2
2
q
5
E  y
 2.5 
 2.70m
2
2
2 gy
2 * 9.81* 2.5
E  E min
3
 y c  y c  1.80 m
2
2
q
3
2
3
yc 
 q  gyc  7.56 m / s
g
b2 = Q/ q2 = 30 / 7.56 = 3.07 m
Weirs & Spillways
To control the elevation of the water
- Functions as a downstream choke control
- Classified as sharp crested or broad crested
depending on critical depth occurrence on
the crest
2
1
2
2
V
V
y1 
 y2 
2g
2g
y2=0
V2  2 gy1  V
2
1
V12
y1 
H
2g
Orifice equation:
V2  2gH
Head on the weir crest
Weirs & Spillways
Assume V1=0
Immediate region of weir crest
Discharge through the element:
Integrate across the head (0 - H):
dQ  VdA  2gH LdH
Total discharge across the weir:
Q  2g L H
1/ 2
2
3/ 2
dH 
2 g LH
3
2
Q  Cd
2 g LH 3 / 2  CLH 3 / 2
3
Coefficient of Discharge
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.
2
Q  Cd
2 g LH 3 / 2  CLH 3 / 2
3
Cd  0.611 0.08H / Z
(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 Kay
Copyright © 1998 by E & FN Spon . All rights reserved.
Discharge Control
Practical Hydraulics by Melvyn Kay
Copyright © 1998 by E & FN Spon . All rights reserved.
WEIRS
WEIRS
FLUMES
Practical Hydraulics by Melvyn Kay
Copyright © 1998 by E & FN Spon . All rights reserved.
Class Exercises:
10-32