Presentation on theme: "12-0 prepared by Ercan Kahya Lecture Notes 3: (Chapter 12) Energy principles in Open-Channel Lecture Notes 3: (Chapter 12) Energy principles in Open-Channel."— Presentation transcript:
12-0 prepared by Ercan Kahya Lecture Notes 3: (Chapter 12) Energy principles in Open-Channel Lecture Notes 3: (Chapter 12) Energy principles in Open-Channel Energy generated at an overfall (Niagara Falls).
12-1 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
12-2 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 sections Weighted mean velocity:
12-3 Specific Energy Assuming 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
12-4 Specific Energy Diagram The specific energy diagram - E s varies linearly with y - E k varies nonlinearly with y - Horizontal sum of the line OD & the curve kk` produces SED - For given E: alternate depths (y 1 & y 2 ) - They are two depths with the same specific energy and conveying the same discharge -E min vs critical depth
12-5 Specific Energy Diagram The specific energy diagram for various discharges - An increase in the required E min yields bigger discharges. - F n : Froude number equals to V square / gD
12-6 Critical Flow Conditions General mathematical formulation for critical flow conditions: - Assume dA/dy = B
12-7 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
12-8 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
12-9 Critical Depth 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 trapezoidal and circular sections For the rectangular cross section:
12-10 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.
12-11 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:
12-12 Critical Slope 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. of the energy & resistance principles as applied to open channels.
12-13 Discharge-Depth Relation for Constant Specific Energy Now assume E o constant, then evaluate Q-y relation: For the condition of the Q max : 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.
12-14 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-y relation Q-y relation for constant specific energy
12-15 Transitions in Channel Beds Consider Consider an open-channel with a small drop ∆z in its bed A small drop in the channel bed (subcritical flow): (a) change in 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.
12-16 Transitions in Channel Beds Consider Consider an abrupt rise ∆z in the open-channel bed Assume that upstream conditions are subcritical & initial E 1 Note that ∆z should be subtracted from E 1 & While TEL unchanged, E reduced
12-17 Transitions in Channel Beds Consider Consider an abrupt rise ∆z in the open-channel bed Assume that upstream conditions are supercritical & initial E 1 Note that ∆z should be subtracted from E 1 & While TEL unchanged, E reduced RESULT : Water depth must rise after the step
Chokes Chokes can only occur when the channel is constricted, but will not occur where the flow area expanded such as drops or expansions. 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. 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 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 m 3 /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 b 2 = Q/ q 2 = 30 / 7.56 = 3.07 m
Weirs & Spillways y 2 =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 Head on the weir crest Orifice equation:
Weirs & Spillways Immediate region of weir crestAssume V 1 =0 Discharge through the element:Integrate across the head (0 - H): Total discharge across the weir:
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 C d 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