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