Presentation on theme: "Total & Specific Energy"— Presentation transcript:
0 Lecture Notes 3: (Chapter 12) Energy principles in Open-Channel Energy generated at an overfall (Niagara Falls).prepared by Ercan Kahya
1 Total & Specific Energy Specific Energy: the energy per unit weight of water measured fromthe channel bottom as a datum► Note that specific energy & total energy are not generally equally.At section 1: Specific Energy Total Energy
2 Total & Specific Energy ► Specific energy varies abruptly as does the channel geometry► Velocity coefficient (α) is used to account nonuniformity of thevelocity 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
3 Specific Energy Diagram (SED) Assuming α equal to 1, it is convenient to express E in terms of Qfor steady flow conditionsf(E, Q, y) = 0Specific 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:whereand
4 Specific Energy Diagram Es varies linearly with yEk varies nonlinearly with yHorizontal sum of the line OD & the curve kk` produces SEDFor given E: alternate depths (y1 & y2)They are two depths with the same specific energy and conveying the same dischargeEmin vs critical depthThe specific energy diagram
5 Specific Energy Diagram An increase in the required Emin yields bigger discharges.Fn : Froude numberequals to V square / gDThe specific energy diagramfor various discharges
6 Critical Flow Conditions General mathematical formulation for critical flow conditions:Assume dA/dy = B
7 Critical Flow Conditions At the critical flow conditions, specific energy is minimum:Then,which can also beexpressed as -->Then,In wide or rectangular section, D = yat critical depth
8 Critical Velocity The general expressions for Used to determine the state of flowCritical state condition:Critical velocity for the general cross section:Velocity head at critical conditions:In wide or rectangular section, D = y
9 Critical Depth For a certain section & given discharge: Critical depth is defined as the depth of flow requiring minimum specific energyThis equation should be solved …For the trapezoidal cross section:Solve this by trial & error …For the rectangularcross section:Critical depth trapezoidal and circular sections
10 Critical EnergyCritical Energy is the energy when the flow is under critical conditions.Recall for any cross section:Then,For wide or rectangular section, D = y
11 Critical SlopeCritical slope is the bed slope of the channel producing critical conditions.► depends discharge; channel geometry; resistance or roughnessFor Chezy equation:Then,For Manning equation:In English unit:For direct computation:
12 Critical SlopeCritical slope is very important in open-channel hydraulics. WHY?The summary given above encompasses much of the important conceptsof the energy & resistance principles as applied to open channels.
13 Discharge-Depth Relation for Constant Specific Energy Now assume Eo constant, then evaluate Q-y relation:For the condition of the Qmax:It reduces toThen substitute this into Q equation at the top:implies that the Qmax is encountered at the critical flow condition for given E.
14 Discharge-Depth Relation for Constant Specific Energy Q-y relationfor constant specific energyFor wide or rectangular section, D = ycan be written asDifferentiating this w.r.t. y and equating to zero:
15 Transitions in Channel Beds Consider an open-channel with a small drop ∆z in its bedAssume that friction losses and minor losses due to drop are negligibleThe method provides a good first approximation of the effects of the transitionFirst step: compare the given conditions to critical conditions to determinethe initial state of flow.A small drop in the channel bed (subcritical flow): (a) change in water levels, and (b) steps for solution.
16 Transitions in Channel Beds Consider an abrupt rise ∆z in the open-channel bedAssume that upstream conditions are subcritical & initial E1Note that ∆z should be subtracted from E1 & While TEL unchanged, E reduced
17 Transitions in Channel Beds Consider an abrupt rise ∆z in the open-channel bedAssume that upstream conditions are supercritical & initial E1Note that ∆z should be subtracted from E1 & While TEL unchanged, E reducedRESULT : Water depth must rise after the step
18 ChokesChokes 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.
19 ChokesFigure 12.16: Rise in a channel bed: (a) a small step-up, (b) a bigger step-up
20 ChokesFigure 12.16: Rise in a channel bed: (c) a still bigger step-up, and (d) changes in the specific energy.
21 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.
22 EXAMPLEA 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.
23 EXAMPLE: s o l u t i o nb2 = Q/ q2 = 30 / 7.56 = 3.07 m
24 Weirs & Spillways y2=0 To control the elevation of the water Functions as a downstream choke controlClassified as sharp crested or broad cresteddepending on critical depth occurrence on the cresty2=0Head on the weir crestOrifice equation:
25 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:
26 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