INTRODUCTIONTheenergyheadatanypositionalongthechannelisthe sum of the vertical distance measured from ahorizontal datum, z, the depth of flow, y, and thekineticenergyheadA
B
CD.Thatsumdefinestheenergy
lineandistermedthetotalenergy,H.
𝐻 = 𝑧 + 𝑦 +𝑉C
2𝑔
Losses will occur for a real fluid between any twosections of the channel. The total energy will notremainconstant.Theenergybalanceisgivenbythisrelation.
𝐻Q = 𝐻C + ℎSwhere;ℎSistheheadloss.Theonlymannerinwhichenergycanbeaddedtoanopenchannelflowsystemisformechanicalpumpingorliftingoftheliquidtotakeplace.
SPECIFICENERGYItisconvenientinopenchannelflowtomeasuretheenergy relative to the bottom of the channel. Itprovides a useful means to analyze complex flowsituations.SpecificenergyisdesignatedasE:Itisthesumoftheflowdepth,𝑦,andkineticenergyhead,A
B
CD.
𝐸 = 𝑦 +𝑉C
2𝑔
RECTANGULARsection:Forarectangularsection,thespecificenergycanbeexpressedasafunctionofthedepth,y.The specific discharge, q, is defined as the totaldischargedividedbythechannelwidth.
𝑞 =𝑄𝑏= 𝑉 ∙ 𝑦
Thespecificenergyforarectangularchannelcanthusbeputintheform:
𝐸 = 𝑦 +𝑞C
2𝑔𝑦C
Figure10.6isshowingE-yrelation.Itisfoundthataspecificdischargerequiresatleastaminimumenergy.This minimum energy is referred to as a criticalenergy,𝐸h.The corresponding depth, 𝑦h , is called the criticaldepth.Ifthespecificenergyisgreaterthan𝐸h,twodepthsarepossible;thosedepthsarereferredtoasalternatedepths.Maximum unit discharge, 𝑞jkl , occurs at criticaldepth.
Thecriticaldepth,𝑦hcanbeevaluatedbysettingthederivativeof𝐸withrespectto𝑦equaltozero.
𝐸 = 𝑦 +𝑞C
2𝑔𝑦C
𝑑𝐸𝑑𝑦 = 1 −
𝑞C
𝑔𝑦o= 0
Masukkan𝑞 = 𝑉𝑦
0 = 1 −𝑉C
𝑔𝑦= 1 − 𝐹𝑟C
Note:Froudenumberinarectangularchannelis:
𝐹𝑟 =𝑉𝑔𝑦
=𝑞𝑔𝑦o
Atcriticaldepth,𝑦h,Froudenumberisequaltoone.
At𝑦 = 𝑦h ⟹ 𝐹𝑟 = 1
𝐹𝑟 = 1 =𝑞
𝑔 𝑦h o
𝑦h =𝑞C
𝑔
Qo
Criticalflowconditions,𝐸h,canconvenientlybeexpressedby:
𝐸h = 𝑦h +𝑞C
2𝑔 𝑦C C
=32𝑦h
OntheE-ycurve;
𝐹𝑟 < 1 Flowisrelativelyslow(tranquil) Subcriticalflow
𝐹𝑟 > 1 Flowisrelativelyrapid(shooting) Supercriticalflow
GENERALIZEDCROSSSECTIONFor a generalized section, the specific energy iswritten in terms of the total discharge,𝑄, and thecross-sectionalarea,𝐴.
𝐸 = 𝑦 +𝑄C
2𝑔𝐴C
The minimum energy condition is obtained bydifferentiatingitwithrespectto𝑦:
𝑑𝐸𝑑𝑦
= 0 = 1 −𝑄C
𝑔𝐴o∙𝑑𝐴𝑑𝑦
Thecorrespondingchangeinareais:
𝑑𝐴 = 𝐵 ∙ 𝑑𝑦Thussettingaboveequationequaltozero:
0 = 1 −𝑄C𝐵𝑔𝐴o
Froudenumbercanwrittenas:
𝐹𝑟 =𝑄C𝐵𝑔𝐴o
=𝑄 𝐴𝑔𝐴 𝐵
=𝑉𝑔𝐴 𝐵
Theratio𝐴 𝐵istermedthehydraulicdepth.𝐴 𝐵isequalto𝑦forarectangularchannel.
CHANNELGEOMETRYAregularsectionisonewhoseshapedoesnotvaryalongthelengthofthechannel.Anirregularsectionwillhavechangesinitsgeometry.
Fortrapezoidal:(P=WP=wettedperimeter)(B=freesurfacewidth)
Forcircularcrosssection:
EXAMPLE1
EXAMPLE2
EXAMPLE3