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**Lecture Notes 3: (Chapter 12) Energy principles in Open-Channel**

Energy generated at an overfall (Niagara Falls). prepared by Ercan Kahya

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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

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**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

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**Specific Energy Diagram (SED)**

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

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**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

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**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

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**Critical Flow Conditions**

General mathematical formulation for critical flow conditions: Assume dA/dy = B

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**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

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**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

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**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

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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

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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:

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Critical Slope Critical slope is very important in open-channel hydraulics. WHY? The summary given above encompasses much of the important concepts of the energy & resistance principles as applied to open channels.

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**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**

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:

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**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.

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**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

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**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

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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.

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

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

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**Enlargements and constructions in channel widths**

(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 channel-supercritical flow.

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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.

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EXAMPLE: s o l u t i o n b2 = Q/ q2 = 30 / 7.56 = 3.07 m

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**Weirs & Spillways y2=0 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 y2=0 Head on the weir crest Orifice equation:

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**Weirs & Spillways Assume V1=0 Immediate region of weir crest**

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

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**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. (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

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**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.

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**Discharge Control Practical Hydraulics by Melvyn Kay**

Copyright © 1998 by E & FN Spon . All rights reserved.

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WEIRS

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WEIRS

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**FLUMES Practical Hydraulics by Melvyn Kay**

Copyright © 1998 by E & FN Spon . All rights reserved.

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Class Exercises:

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