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Chapitre 4

Hydraulique en charge

O. Thual, 12 juin 2021

Sommaire1 Transition vers la turbulence . . . . . . . . . . . . .2

1.1 Pertes de charge dans les conduites en charge . . .2

1.2Ecoulements laminaires ou turbulents . . . . . . . .3

1.3 Nombre de Reynolds critique . . . . . . . . . . . . .4

2 Calcul des pertes de charge . . . . . . . . . . . . . .

5

2.1 Formule de Colebrook . . . . . . . . . . . . . . . . .5

2.2 Formules explicites du coecient de frottement . . .7

3 Pertes de charge singulieres . . . . . . . . . . . . . .

8

3.1 Coecient de perte de charge singuliere . . . . . . .8

3.2 Longueur equivalente . . . . . . . . . . . . . . . . . .10 1

2Chapitre 4. Hydraulique en chargeIntroduction

La plupart des ecoulements en charge sont contenus dans les conduites cir- culaires, ce qui explique que seul ce cas, facilement generalisable, est aborde ici. Lorsque le debit est petit, l'ecoulement est laminaire et le prol de vitesse est parabolique. Les pertes de charge sont alors proportionnelles a la vitesse, comme dans un milieu poreux, et le coecient de frottement est inversement proportionnel au nombre de Reynolds. Lorsque ce nombre de Reynolds depasse une valeur critique, l'ecoulement devient turbulent, ce qui se traduit par une augmentation du coecient de frottement, comme on peut le voir sur le dia- gramme de Moody. La formule implicite de Colebrook, qui rend compte des valeurs experimentales de ce diagramme, peut ^etre approximee par des for- mules donnant explicitement la valeur du coecient de frottement. Les pertes de charge a travers des variations brusques rencontrees dans l'ecoulement, comme par exemple des elargissements brusques, des coudes ou des vannes, sont decrites a l'aide de coecients de pertes de charge singulieres. Ces singula- rites peuvent ^etre representees par une conduite dont la \longueur equivalente" genere les m^emes pertes de charge.

1 Transition vers la turbulence

1.1 Pertes de charge dans les conduites en chargeFigure4.1 {Perte de chargehfdans une conduite circulaire de longueurL.

La charge hydraulique des ecoulements en charge (voir chapitre 2), est denie par la relation

H=Z+Pg

+V22g;(4.1) ouZest la cote verticale de l'axe de la conduite,Pla pression sur cet axe et Vla vitesse moyenne dans la section. On peut visualiser la charge hydraulique comme etant la cote de la surface libre d'un tube de Pitot (gure4.1) augmente dePa=(g) ouPaest la pression atmospherique. Les frottements sur la paroi d'une conduite de longueurLengendrent une FORMULAIRE13la formule de Colebrook ou de Hazen-Williams, respectivement : h sL eq=f(r;Re)V22g DouhsL eq= 10;675QC HW

1;8521D

4;87:

14Chapitre 4. Hydraulique en chargeEXERCICES

EXERCICE 4.1Rendements d'un barrage hydroelectrique On considere l'ecoulement en charge amenant l'eau d'une retenue de cote de surface libreZga une retenue de cote de surface libreZvtelle que la hauteur brute est egale ahbrut=ZgZv= 900 m (voir gure4.11).L n

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n

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.11 {Barrage hydroelectriqueUet ses galeries et conduites forcees.

On suppose que les conduites, de sections circulaires, sont une galerie souter- raine en betonCbde longueurLb= 7 km, de diametreDb= 7 m et de rugosite absolueb= 2;8 mm, suivie d'une conduite forcee en acierCnde longueur L n= 1;5 km, de diametreDn= 3 m et de rugosite absoluen= 2;4 mm. La galerie et la conduite forcee sont reliees par un retrecissement brusque, suivi de deux coudes de rayon de courburec= 5 m et d'angle de deviation '==3. Une cheminee d'equilibre met en contact l'eau de la conduiteCb avec l'atmosphere au moyen d'une conduite verticale susamment haute pour eviter tout debordement. La conduiteCnalimente une usine hydroelectrique Udont le rendement des turbines estturb= 0;93 et le rendement des al- ternateurs et transformateurs pour convertir l'energie mecanique en energie electrique estmec= 0;92. On denit le rendement globala partir de la puissance electrique recuperable rec(Q) = g hbrutQouQest le debit. 1) On supp oseque la vitesse dans la cond uiteen acier CnestVn= 10 m/s.

Calculer le debitQ10dans les conduitesCnetCb.

2)

Calculer la vitesse Vbdans la conduiteCb.

3) Calculer le co ecientde frottemen tfndans la conduiteCn. 4) Calculer le co ecientde frottemen tfbdans la conduiteCb. 5) En d eduirela v aleurde la som medes p ertesde c hargelin eiqueshfdans les conduites. 6) Calculer la somme des p ertesde c hargesinguli ereshsentre les retenues amont et aval. 7) On d enitla hauteur nette comme etantla hauteur du r eservoird'altitude qui produirait, pour un m^eme debit, la m^eme charge a l'amont de l'usine Usi les pertes de charge dans les conduites etaient nulles. Calculer la hauteur nettehnetet le rendement des conduitescond=hnet=hbrutpour le debitQ10. 8) En d eduirele rendemen tglobal 10et la puissance electrique rec(Q10)

EXERCICES15delivree pour le debitQ10.

9) Mon trerque l'on p eut ecrire,dans le cas des r egimesrugueux, rec(Q) = turbmecg(hbrutQ2)Qouest coecient que l'on calculera en negligeant les pertes de charge singulieres. En deduire la puissance maxi- male met le debitQmcorrespondant. Quel est, dans ce cas, le rendement globalm? 10) Que deviennen tles co ecientsde frottemen tfbetfnlorsque le debit est divise par dix, cent ou mille.Corrige 4.1Rendements d'un barrage hydroelectrique

1)Le debit estQ10=VnAn= 71 m3/s avecAn= D2n=4 = 7;1 m2.2)La

vitesseVbdans la conduiteCben beton estVb=Q10=Ab= 1;8 m/s avec A b= D2b=4 = 38;5 m2.3)Le nombre de Reynolds dans la conduiteCbest Re b=VbDb== 1;3 107. Comme la rugosite relative de la conduiteCbest r b= 4104, on lit sur le diagramme de Moody ou en resolvant l'equation implicite de Colebrook quefb= 0;016.4)Le nombre de Reynolds dans la conduiteCnestRen=VnDn== 3;0 107. Comme la rugosite relative de la conduiteCnestrn= 8104, on lit sur le diagramme de Moody ou en resolvant l'equation implicite de Colebrook quefn= 0;019.5)Les pertes de charge lineiques sonthfb=LbfbV2b=(2g Db) = 2;7 m pour la conduiteCb ethfn=LnfnV2n=(2g Dn) = 47;4 m pour la conduiteCn. La somme des pertes de charge lineiques vauthf=hfb+hfn= 50 m.6)Les coecients de pertes de charge singulieres sont :Kbn= 0;5[1(Dn=Db)2] = 0;41 pour le retrecissemenent brusque etKc=2'c h

0;131 + 1;847 (2c=Dn)3;5i

0;11 pour chacun des coudes. On en deduit les pertes de chargehsbn=

K sbnV2b=(2g) = 0;07 m et deux foishsc=KscV2n=(2g) = 0;54 m. La somme des pertes de charge singulieres esths=hsbn+ 2hsc= 1;1 m.7)La perte de charge totale est H=hf+hs= 51 m si bien que la hauteur nette esthnet=hbrutH= 849 m. Le rendement d^u aux conduites est donccond= 0;94, ce qui signie que 16% de la hauteur brute a ete consommee en perte de charge dans les conduites.8)Le rendement global est =condturbmec= 0;8. La puissance electrique recuperable correspondant au debitQ10= 71 m3/s est rec(Q10) = 500 MW.9)On peut ecrire la perte de charge sous la forme H=Q2avec= 8(fbLb=D5b+fnLn=D5n)(2g) en negligeant les pertes de charges singulieres. On obtient= 1;0102si on suppose que l'ecoulement reste rugueux, avecfb= 0;016 etfn= 0;019, ce que l'on verie a posteriori. La puissance rec(Q) =turbmecg[hbrutQ2]Q est maximale pourhnet=hbrut= 2=3 (rendement des conduites de 66 %) et vaut donc m= 862 MW avecQm= 171 m3/s etVm= 24 m/s. Le rendement global vaut alorsm= 0;57.10)Pour les debits consideres, les nombres de Reynolds sont respectivementReb21;3f107;106;105;104gpour la conduiteCbetRen23f107;106;105;104gpour la conduiteCn. Les coe- cients de perte de charge prennent alors les valeursfb2 f16;16;19;29g102 etfn2 f19;19;20;25g102.

16Chapitre 4. Hydraulique en chargeNOTATIONS

AAire d'une section (m2)

C

HWCoecient de Hazen-Williams ()

D

HDiametre hydraulique (m)

DDiametre (m)

fCoecient de frottement () f lamiCoecient de frottement du regime laminaire () f lissCoecient de frottement du regime lisse () f rugCoecient de frottement du regime rugueux () gGravite (m.s2)

HCharge hydraulique (m)

h fPerte de charge dans une conduite (m) h sPerte de charge singuliere (m)

KCoecient de perte de charge singuliere ()

K pConductivite hydraulique (m.s1)

LLongueur de la conduite (m)

L eqLongueur equivalente (m)

PPression (Pa)

P aPression atmospherique (Pa)

QDebit (m3.s1)

rRugosite relative ()

ReNombre de Reynolds ()

sCoordonnee curviligne (m) S fPente de frottement () uProl de vitesse (m.s1)

VVitesse moyenne (m.s1)

ZCoordonnee verticale (m)

Constante ()

Rugosite absolue (m)

rendement global () turbrendement d'une turbine () mecrendement alternateur et transformateur () condrendement des conduites ()

Viscosite cinematique (m2.s1)

recPuissance recuperable (W)

Masse volumique de l'eau (kg.m3)

'Angle de deviation (rad)

0Angle de fermeture (rad)

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