Hydraulique en charge - INP Toulouse

Chapitre 4

Hydraulique en charge

O. Thual, 12 juin 2021

Sommaire

1 Transition vers la turbulence . . . . . . . . . . . . . 2

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

1.2 Ecoulements 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 coefficient de frottement . . . 7

3 Pertes de charge singulieres . . . . . . . . . . . . . . 8

3.1 Coefficient de perte de charge singuliere . . . . . . . 8

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

1

2 Chapitre 4. Hydraulique en charge

Introduction

La plupart des ecoulements en charge sont contenus dans les conduites cir-culaires, ce qui explique que seul ce cas, facilement generalisable, est abordeici. Lorsque le debit est petit, l’ecoulement est laminaire et le profil de vitesseest parabolique. Les pertes de charge sont alors proportionnelles a la vitesse,comme dans un milieu poreux, et le coefficient de frottement est inversementproportionnel au nombre de Reynolds. Lorsque ce nombre de Reynolds depasseune valeur critique, l’ecoulement devient turbulent, ce qui se traduit par uneaugmentation du coefficient de frottement, comme on peut le voir sur le dia-gramme de Moody. La formule implicite de Colebrook, qui rend compte desvaleurs experimentales de ce diagramme, peut etre approximee par des for-mules donnant explicitement la valeur du coefficient de frottement. Les pertesde 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 coefficients de pertes de charge singulieres. Ces singula-rites peuvent etre representees par une conduite dont la “longueur equivalente”genere les memes pertes de charge.

1 Transition vers la turbulence

1.1 Pertes de charge dans les conduites en charge

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Figure 4.1 – Perte de charge hf dans une conduite circulaire de longueur L.

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

H = Z +P

ρ g+V 2

2 g, (4.1)

ou Z est la cote verticale de l’axe de la conduite, P la pression sur cet axe etV la vitesse moyenne dans la section. On peut visualiser la charge hydrauliquecomme etant la cote de la surface libre d’un tube de Pitot (figure 4.1) augmentede Pa/(ρ g) ou Pa est la pression atmospherique.

Les frottements sur la paroi d’une conduite de longueur L engendrent une

Transition vers la turbulence 3

perte de charge hf et l’on definit la “pente de frottement” par la relation

Sf = −dHds

=hfL. (4.2)

La relation de Darcy-Weisbach (chapitre 3) relie cette pente de frottement ala vitesse V et au diametre hydraulique DH , par

Sf = f(r,Re)V 2

2 g DH(4.3)

ou le coefficient de frottement f depend de la rugosite r = ε/DH et du nombrede Reynolds Re = V DH/ν, avec ε la rugosite absolue (voir table au chapitre 3)et ν la viscosite cinematique.

Dans la suite de ce chapitre, on se place dans le cas des conduites circulaires,ce qui entraine que DH = D ou D est le diametre de la section.

1.2 Ecoulements laminaires ou turbulents

Les equations de Navier-Stokes incompressibles, qui traduisent la conservationde la masse et de la quantite de mouvement, decrivent des solutions station-naires pour l’ecoulement d’un fluide newtonien (comme l’eau) de viscositecinematique ν dans une conduite circulaire de diametre D en presence d’ungradient de charge dH/ds < 0 (voir par exemple [?]). La vitesse (figure 4.2),parallele a l’axe du cylindre, ne depend que de la distance r au centre et s’ecrit :

u(r) = −KpdH

ds

(2− 8

r2

D2

)avec Kp =

D2 g

32 ν. (4.4)

L’ecoulement est laminaire dans la mesure ou les filets d’eau, ou lames d’eau,sont reguliers.

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Figure 4.2 – Profil de vitesse u(r) pour un ecoulement laminaire dans uneconduite circulaire.

On calcule alors la vitesse moyenne V =[∫D/2

0 u(r)(2π r) dr]/(πD2/4), ce

qui conduit a la relation

V = −KpdH

ds. (4.5)

On peut interpreter Kp comme une conductivite hydraulique (voir chapitre 2)dans la mesure ou la vitesse est proportionnelle a la perte de charge, comme


dans un milieu poreux. Cette propriete permet de comprendre la loi de Darcyqui decrit la superposition d’un grand nombre d’ecoulements laminaires entreles pores d’un milieu poreux.

On peut aussi ecrire cette expression sous la forme d’une relation de Darcy-Weisbach

Sf = flami(Re)V 2

2 g Davec flami(Re) =

64

Re, (4.6)

ou Re = V D/ν est le nombre de Reynolds. Le coefficient f(Re) est inverse-ment proportionnel au nombre de Reynolds, ce qui ce traduit par une droitede pente -1 dans le diagramme de Moody dans la mesure ou les axes sonttraces en echelles logarithmiques (figure 4.4).

1.3 Nombre de Reynolds critique

L’experience de Reynolds (figure 4.3) consiste a documenter les valeurs desparametres D, V et ν pour lesquels l’ecoulement laminaire devient turbulent,ce qui se traduit par des trajectoires desordonnees des particules fluides quel’on peut mettre en evidence a l’aide d’un colorant.

Figure 4.3 – Experience de Reynolds (1883).

La transition du regime laminaire au regime turbulent s’observe lorsque lenombre de Reynolds Re = V D/ν depasse la valeur critique Rec ∼ 2 300.Cette transition se traduit par une augmentation brusque du coefficient defrottement f , comme on peut le constater sur le diagramme de Moody (fi-gures 4.4 ou 4.10). Des calculs theoriques complexes (theorie des instabilites)permettent de determiner ce seuil de transition.

Lorsque le nombre de Reynolds est augmente au-dela du seuil critique Rec, lecoefficient de frottement decroit, mais bien moins vite que pour le cas laminaire(figure 4.4). Cette decroissance suit tout d’abord une courbe fliss(Re) qui nedepend que de Re. On dit que le regime de l’ecoulement est lisse. Lorsque lenombre de Reynolds continue d’etre augmente, le coefficient f(r,Re) traverseune zone de transition pour rejoindre une courbe frug(r) qui ne depend quede la rugosite relative r. On dit que le regime de l’ecoulement est rugueux. La

Calcul des pertes de charge 5

lecture des valeurs de f est plus ou moins aisee selon les representations dudiagramme de Moody (voir figure 4.10 pour un graphique bien resolu).

2 Calcul des pertes de charge

2.1 Formule de Colebrook

La relation de Darcy-Weisbach appliquee aux ecoulements en charge dans desconduites circulaires, pour lesquels le diametre hydraulique DH est egal audiametre de la conduite D, s’ecrit

−dHds

=hfL

= Sf = f(r,Re)V 2

2 g D(4.7)

ou Sf est la pente de frottement, r = ε/D la rugosite relative, Re = V D/νle nombre de Reynolds et V la vitesse debitante. Le coefficient de frottementf(r,Re) se determine a l’aide du diagramme de Moody des conduites circu-laires (figures 4.4 ou 4.10) ou, lorsque Re > Rec, a partir de la “formule deColebrook” suivante

1√f

= −2 log10

(r

3, 71+

2, 51

Re√f

). (4.8)

Il s’agit d’une equation implicite de la forme f = F(f) que l’on peut resoudrede maniere iterative en choisissant une valeur raisonnable de f0 puis en cal-culant f1 = F(f0), f2 = F(f1)... jusqu’a ce que la precision soit satisfaisante(troisieme decimale).

La formule de Colebrook permet de bien decrire les regimes lisses et rugueuxdes ecoulements. Si Re n’est pas trop grand et r suffisamment petit, le regimeest lisse et l’on peut determiner fliss(Re) a l’aide de l’equation implicite

1√f liss

= −2 log10

(2, 51

Re√f liss

). (4.9)

A contrario, pour une rugosite r donnee, il existe une valeur de Re (voir courbenoire sur la figure 4.4) au-dela de laquelle le regime est rugueux, si bien quefrug(r) est determine par la relation :

1√f rug

= −2 log10

(r

3, 71

). (4.10)

Dans la mesure ou frug ne depend pas de la vitesse V mais uniquement descaracteristiques de la conduite, la dependance de la pente de frottement avecle debit Q = (πD2/4)V est quadratique :

Sf = frug(r)V 2

2 g D=

8

g π2frug(r)

Q2

D5. (4.11)

Pour les applications pratiques, plutot que de resoudre d’entree l’equation im-plicite pour f , on pourra d’abord supposer que l’ecoulement est rugueux, utili-ser cette formule et verifier a posteriori si l’hypothese est valide. Il existe aussides approximations qui permettent d’expliciter f en fonction des parametresdu probleme. Elles sont decrites ci-dessous.


f

Re

Laminaire

Turb

ulen

t

r=

.0001

r=

.01

r=

.001

104

105

106

107

.01

.1

42 8

Lisse

Rugueux

fliss(Re)

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frug(r)<latexit sha1_base64="q0dJ7teN5HkyLsNA9e/POp7517k=">AAAC8nicjVLLSsNAFD3GV31XXXYTFEE3JXVj3RXduKxgrfhAkjitwTQJk4lQSv/CnSiIW3/IP1CXfoFnphF84OOGJGfOvefO3HvHS8IgVY7zOGQNj4yOjRcmJqemZ2bnivML+2mcSV80/DiM5YHnpiIMItFQgQrFQSKF2/FC0fQutrW/eSlkGsTRnuom4qTjtqOgFfiuInXYOu3JrN1flWunxWWn7Bizv4NKDpZrpdct0Opx8QXHOEMMHxk6EIigiEO4SPkcoQIHCbkT9MhJosD4BfqYpDZjlGCES/aC3zZXRzkbca1zpkbtc5eQr6TSxgo1MeMksd7NNv7MZNbsT7l7Jqc+W5d/L8/VIatwTvYv3Xvkf3W6FoUWqqaGgDUlhtHV+XmWzHRFn9z+UJVihoScxmf0S2LfKN/7bBtNamrXvXWN/8lEalav/Tw2w/Ov1ekTDHrNufAaVL4O/TtorJc3y84ur0MVAyughCWscugbqGEHdTS4SYQr3ODWyqxr6866H4RaQ7lmEZ/MengDvlac6w==</latexit><latexit sha1_base64="iea2Cgm+wXEHCc8rE9xKRSf+r3A=">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</latexit><latexit sha1_base64="iea2Cgm+wXEHCc8rE9xKRSf+r3A=">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</latexit>

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flami(Re)

Figure 4.4 – Diagramme de Moody pour les conduites circulaires.

Calcul des pertes de charge 7

2.2 Formules explicites du coefficient de frottement

Parmi les nombreuses approximations explicites de la formule de Colebrook,la formule de Haaland (1983) est sans doute la plus simple :

1√f

= −1, 8 log10

[(r

3, 7

)1,11

+6, 9

Re

]. (4.12)

Les ecarts a la formule de Colebrook, inferieurs au pour cent, sont en-deca deserreurs des donnees experimentales qui justifient ces valeurs.

Une autre approximation de la formule de Colebrook est donnee par la formulede Hazen-Williams (table 4.1). Cette formule, tres pratique pour les applica-tions, necessite de disposer de tables de coefficients de Hazen-Williams CHW ,caracteristiques des materiaux utilises. Ces tables sont alternatives aux tablesde rugosites absolues.

Materiaux CHWPolytethylene 150

Cuivre 130-140

Acier 110-120

Fonte 100-140

Beton 90-130

Sf = 10, 675

(Q

CHW

)1,852 1

D4,87. (4.13)

Table 4.1 – Valeurs de coefficients CHW et formule de Hazen-Williams.

Pour une conduite donnee, la formule de Hazen-Williams permet de tracerfacilement la dependance de la pente de frottement Sf avec le debit, commerepresente sur la figure 4.5. Ces courbes permettent de se rendre compte queles pertes de charge augmentent avec le debit et sont d’autant plus grandesque le diametre de la conduite est petit.

0 10.2 0.4 0.6 0.80.1 0.3 0.5 0.7 0.90

0.1

0.02

0.04

0.06

0.08

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=.2

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D=

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D=

.4

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D = .5<latexit sha1_base64="1gQzVjvhzyyEa6hreizYwwl0eYE=">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</latexit><latexit sha1_base64="RlQXa/2VS0E0k9FpGTXs7WmVlvU=">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</latexit><latexit sha1_base64="RlQXa/2VS0E0k9FpGTXs7WmVlvU=">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</latexit>

D = .6<latexit sha1_base64="lAQ0owpRzlCWklr2siu7NNVbNM8=">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</latexit><latexit sha1_base64="Yd3xtZfsC1pJSgRql6Sq1LNCWt0=">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</latexit><latexit sha1_base64="Yd3xtZfsC1pJSgRql6Sq1LNCWt0=">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</latexit>

D = .7<latexit sha1_base64="MgZvR/QqDRlGqqAFbw5/vhKhPjc=">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</latexit><latexit sha1_base64="1H7fqOBxzLQhJ3RFK2ZsbKlSr1s=">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</latexit><latexit sha1_base64="1H7fqOBxzLQhJ3RFK2ZsbKlSr1s=">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</latexit>

D = .8<latexit sha1_base64="RqOc9HVV9WTPYawD06gSNkrQIXc=">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</latexit><latexit sha1_base64="VOUmPmDDggVWTuT9cev6DILryVM=">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</latexit><latexit sha1_base64="VOUmPmDDggVWTuT9cev6DILryVM=">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</latexit>

D = .9<latexit sha1_base64="dtNYfjXEBnNB2hhjAVil79mpjnU=">AAAC7HicjVLLSsNAFD2Nr1pfVZdugkVwFVI3tguxoAuXVYwt1CLJdNqGpklIJkIp/oI7URC3foMr/8M/UL/CO9MU1OLjhrRnzj3nztw7cULPjYVpvmS0qemZ2bnsfG5hcWl5Jb+6dhYHScS4xQIviOqOHXPP9bklXOHxehhxu+94vOb0DmS+dsmj2A38UzEIebNvd3y37TJbSOpwzyhf5AumYarQJ0ExBYX956cTUFSD/DvO0UIAhgR9cPgQhD3YiOlpoAgTIXFNDImLCLkqz3GFHHkTUnFS2MT26LdDq0bK+rSWNWPlZrSLR29ETh1b5AlIFxGWu+kqn6jKkv2p9lDVlGcb0L+T1uoTK9Al9i/fWPlfn+xFoI2S6sGlnkLFyO5YWiVRU5En1z91JahCSJzELcpHhJlyjuesK0+sepeztVX+VSklK9cs1SZ4+7U7eYLRrOle6DMofr/0SWDtGGXDPDYLlRJGkcUGNrFNl76LCo5QhUWbdHGNW9xpgXaj3WsPI6mWST3r+BLa4weWx5pB</latexit><latexit sha1_base64="C+EnO7fhwoJvOpC+0U1KMDbUfjk=">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</latexit><latexit sha1_base64="C+EnO7fhwoJvOpC+0U1KMDbUfjk=">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</latexit>

Q<latexit sha1_base64="0a9wXSMjm19ntwgssd7Txdgzi+w=">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</latexit><latexit sha1_base64="ptvFYh1gGUwsN15FfYourf0CsE4=">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</latexit><latexit sha1_base64="ptvFYh1gGUwsN15FfYourf0CsE4=">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</latexit>

Sf<latexit sha1_base64="7eq6XDDtdaaB0ZAtmZZ3A8jyOpU=">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</latexit><latexit sha1_base64="uzGU+Bimc7zmUFlMHnfSg7igbAM=">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</latexit><latexit sha1_base64="uzGU+Bimc7zmUFlMHnfSg7igbAM=">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</latexit>

Figure 4.5 – Pente de frottement Sf en fonction du debit Q (m3.s−1) pourdifferentes valeurs du diametre D (m) a partir de la formule de Hazen-Williamspour CHW = 140.


3 Pertes de charge singulieres

3.1 Coefficient de perte de charge singuliere

La pente de frottement Sf = −dH/ds decrit des pertes de charge regulierementreparties le long d’une conduite, induisant une perte de charge hf sur unelongueur L avec Sf = hf/L. Ces pertes de charge lineiques sont repartiessur de grandes longueurs. Elles sont completees par des pertes de charge sin-gulieres qui operent sur de courtes distances : elargissements ou retrecissementsbrusques, vannes, coudes, tes, etc. Une analyse dimensionnelle simple permetd’exprimer une perte de charge singuliere hs (m) en fonction de la vitesse V(m.s−1) en amont de singularite et la gravite g (m.s−2), a travers la relation :

hs = KV 2

2 g. (4.14)

Les valeurs du coefficient de perte de charge singuliere K (sans dimension)doivent en general etre recherchees dans des tables ou des abaques fournis parles fabricants de materiel (voir des exemples figure 4.7).

D2

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Figure 4.6 – Coefficient de perte de charge singuliere K pour unretrecissement ou un elargissement brusque.

Dans le cas d’un retrecissement brusque (D1 > D2) ou d’un elargissementbrusque (D1 < D2) pour des conduites circulaires, les formules suivantes sontproposees et representees sur la figure 4.6 :

K =

0, 5

[1−

(D2D1

)2]si D1 > D2 ,[

1−(D2D1

)−2]2

si D1 < D2 .(4.15)

Le coefficient de perte de charge singuliere dans un coude depend de l’anglede deviation ϕ de l’ecoulement et du rayon de courbure ρc comme representesur la figure 4.9 ou a travers la relation

K =2ϕ

π

[0, 131 + 1, 847

(2 ρcD

)−3,5]. (4.16)

Le coefficient de perte de charge singuliere dans une derivation depend del’angle de deviation ϕ comme indique dans la figure 4.8.

Pertes de charge singulieres 9

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Figure 4.7 – Exemple d’abaques de coefficients de perte de charge singuliereK pour differentes vannes proposees par un fabricant. Sources [?].

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ϕ π/12 π/6 π/4 3π/6 π/2

K 0,1 0,3 0,5 0,7 1,3

Figure 4.8 – Coefficient de perte de charge singuliere pour une derivationd’angle ϕ.


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⇡K

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⇢cD

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Figure 4.9 – Coefficient de perte de charge singuliere pour un coude de rayonρc et d’angle ϕ.

3.2 Longueur equivalente

La longueur equivalente Leq associee a une singularite hydraulique est definiecomme la longueur de la conduite de diametre D qui produirait une perte decharge hf egale a la perte de charge singuliere hs pour une vitesse V donnee.En utilisant les equations (4.7) et (4.14), respectivement

hsLeq

= f(r,Re)V 2

2 g Det hs = K

V 2

2 g, (4.17)

la longueur equivalente est donc definie par

Leq =KD

f(r,Re). (4.18)

Cette definition est utile dans le cas des regimes rugueux ou f(r) ne dependpas du Reynolds, ce qui permet d’affirmer que Leq ne depend pas du debit.

La notion de longueur equivalente peut etre egalement definie avec la formulede Hazen-William (4.13) ce qui conduit a

hsLeq

= 10, 675

(Q

CHW

)1,852 1

D4,87. (4.19)

FORMULAIRE

Charge hydraulique

La charge hydraulique des ecoulements en charge est definie en fonction dela cote Z du centre de la section, de la pression P au centre et de la vitessedebitante V dans la conduite :

H =P

ρ g+ Z +

V 2

2 g.

FORMULAIRE 11

2 4 6 8 2 4 6 8

2 4 6 8 2 4 6 8

2 4 6 8 10

310 4

10 510

610 7

810

0.01

0.012

0.100

0.090

0.080

0.070

0.060

0.050

0.040

0.035

0.030

0.025

0.018

0.016

0.014

0.020

g ud L

hf

2

2

⋅=

ν du

=R

e

pipe friction chartapplicable to circular pipes running full

© G

lasgow C

ollege of Nautical S

tudies Faculty of Engineering

GC

NS

runs distance learning courses for the E

ngineering Council G

raduate Diplom

a.

website: w

ww

.glasgow-nautical.ac.uk

e-mail: engineering@

glasgow-nautical.ac.uk

Figure 4.10 – Diagramme de Moody pour les conduites circulaires.


Formule de Darcy-Weisbach

La pente de charge Sf est le quotient de la perte de charge hf sur la longueurL d’une portion de conduite. C’est donc la perte de charge lineique −dH/ds.Elle s’exprime en fonction de la vitesse V , du diametre D de la conduite et ducoefficient de frottement f par la formule de Darcy-Weisbach :

−dHds

=hfL

= Sf = f(r,Re)V 2

2 g D.

Formule de Colebrook

Le coefficient de frottement f(r,Re) depend de la rugosite relative r = ε/D etdu nombre de Reynolds Re = V D/ν a travers la formule de Colebrook :

1√f

= −2 log10

(r

3, 71+

2, 51

Re√f

).

Le diagramme de Moody, qui represente graphiquement f(r,Re), fait appa-raitre un regime laminaire avec flami = 64/Re, un regime lisse avec fliss(Re)et un regime rugueux frug(r).

Formule de Hazen-Williams

La formule de Hazen-Williams, moins precise mais plus pratique, est souventutilisee dans les applications, a condition de disposer des valeurs des coeffi-cients CHW :

hfL

= 10, 675

(Q

CHW

)1,852 1

D4,87H

.

Pertes de charge singulieres

Le coefficient de pertes de charge singulieres est defini par la formule suivante,dans laquelle V est la vitesse en amont de la singularite :

hs = KV 2

2 g.

Longueur equivalente

Une singularite peut etre remplacee par une conduite de meme diametre dontla longueur equivalente produit la meme perte de charge singuliere en utilisant

FORMULAIRE 13

la formule de Colebrook ou de Hazen-Williams, respectivement :

hsLeq

= f(r,Re)V 2

2 g Dou

hsLeq

= 10, 675

(Q

CHW

)1,852 1

D4,87.


EXERCICES

EXERCICE 4.1 Rendements d’un barrage hydroelectrique

On considere l’ecoulement en charge amenant l’eau d’une retenue de cote desurface libre Zg a une retenue de cote de surface libre Zv telle que la hauteurbrute est egale a hbrut = Zg − Zv = 900 m (voir figure 4.11).

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Figure 4.11 – Barrage hydroelectrique U et ses galeries et conduites forcees.

On suppose que les conduites, de sections circulaires, sont une galerie souter-raine en beton Cb de longueur Lb = 7 km, de diametre Db = 7 m et de rugositeabsolue εb = 2, 8 mm, suivie d’une conduite forcee en acier Cn de longueurLn = 1, 5 km, de diametre Dn = 3 m et de rugosite absolue εn = 2, 4 mm.La galerie et la conduite forcee sont reliees par un retrecissement brusque,suivi de deux coudes de rayon de courbure ρc = 5 m et d’angle de deviationϕ = π/3. Une cheminee d’equilibre met en contact l’eau de la conduite Cbavec l’atmosphere au moyen d’une conduite verticale suffisamment haute poureviter tout debordement. La conduite Cn alimente une usine hydroelectriqueU dont le rendement des turbines est ηturb = 0, 93 et le rendement des al-ternateurs et transformateurs pour convertir l’energie mecanique en energieelectrique est ηmec = 0, 92. On definit le rendement global η a partir de lapuissance electrique recuperable Πrec(Q) = η ρ g hbrutQ ou Q est le debit.

1) On suppose que la vitesse dans la conduite en acier Cn est Vn = 10 m/s.Calculer le debit Q10 dans les conduites Cn et Cb.

2) Calculer la vitesse Vb dans la conduite Cb.3) Calculer le coefficient de frottement fn dans la conduite Cn.

4) Calculer le coefficient de frottement fb dans la conduite Cb.5) En deduire la valeur de la somme des pertes de charge lineiques hf dans

les conduites.

6) Calculer la somme des pertes de charge singulieres hs entre les retenuesamont et aval.

7) On definit la hauteur nette comme etant la hauteur du reservoir d’altitudequi produirait, pour un meme debit, la meme charge a l’amont de l’usineU si les pertes de charge dans les conduites etaient nulles. Calculer lahauteur nette hnet et le rendement des conduites ηcond = hnet/hbrut pourle debit Q10.

8) En deduire le rendement global η10 et la puissance electrique Πrec(Q10)

EXERCICES 15

delivree pour le debit Q10.

9) Montrer que l’on peut ecrire, dans le cas des regimes rugueux, Πrec(Q) =ηturb ηmec ρ g (hbrut − αQ2)Q ou α est coefficient que l’on calculera ennegligeant les pertes de charge singulieres. En deduire la puissance maxi-male Πm et le debit Qm correspondant. Quel est, dans ce cas, le rendementglobal ηm ?

10) Que deviennent les coefficients de frottement fb et fn lorsque le debit estdivise par dix, cent ou mille.

Corrige 4.1 Rendements d’un barrage hydroelectrique

1)Le debit est Q10 = VnAn = 71 m3/s avec An = πD2n/4 = 7, 1 m2. 2)La

vitesse Vb dans la conduite Cb en beton est Vb = Q10/Ab = 1, 8 m/s avecAb = πD2

b/4 = 38, 5 m2. 3)Le nombre de Reynolds dans la conduite Cb estReb = VbDb/ν = 1, 3 107. Comme la rugosite relative de la conduite Cb estrb = 4 10−4, on lit sur le diagramme de Moody ou en resolvant l’equationimplicite de Colebrook que fb = 0, 016. 4)Le nombre de Reynolds dans laconduite Cn est Ren = VnDn/ν = 3, 0 107. Comme la rugosite relative dela conduite Cn est rn = 8 10−4, on lit sur le diagramme de Moody ou enresolvant l’equation implicite de Colebrook que fn = 0, 019. 5)Les pertes decharge lineiques sont hfb = Lb fb V

2b /(2 g Db) = 2, 7 m pour la conduite Cb

et hfn = Ln fn V2n /(2 g Dn) = 47, 4 m pour la conduite Cn. La somme des

pertes de charge lineiques vaut hf = hfb + hfn = 50 m. 6)Les coefficients depertes de charge singulieres sont : Kbn = 0, 5 [1 − (Dn/Db)

2] = 0, 41 pour

le retrecissemenent brusque et Kc = 2ϕc

π

[0, 131 + 1, 847 (2 ρc/Dn)−3,5

]=

0, 11 pour chacun des coudes. On en deduit les pertes de charge hsbn =Ksbn V

2b /(2 g) = 0, 07 m et deux fois hsc = Ksc V

2n /(2 g) = 0, 54 m. La

somme des pertes de charge singulieres est hs = hsbn + 2hsc = 1, 1 m. 7)Laperte de charge totale est ∆H = hf + hs = 51 m si bien que la hauteurnette est hnet = hbrut − ∆H = 849 m. Le rendement du aux conduitesest donc ηcond = 0, 94, ce qui signifie que 16% de la hauteur brute a eteconsommee en perte de charge dans les conduites. 8)Le rendement global estη = ηcond ηturb ηmec = 0, 8. La puissance electrique recuperable correspondantau debit Q10 = 71 m3/s est Πrec(Q10) = 500 MW. 9)On peut ecrire la pertede charge sous la forme ∆H = αQ2 avec α = 8 (fb Lb/D

5b + fn Ln/D

5n)(π2 g)

en negligeant les pertes de charges singulieres. On obtient α = 1, 0 10−2 si onsuppose que l’ecoulement reste rugueux, avec fb = 0, 016 et fn = 0, 019, ce quel’on verifie a posteriori. La puissance Πrec(Q) = ηturb ηmec ρ g [hbrut − αQ2]Qest maximale pour hnet/hbrut = 2/3 (rendement des conduites de 66 %) etvaut donc Πm = 862 MW avec Qm = 171 m3/s et Vm = 24 m/s. Lerendement global vaut alors ηm = 0, 57. 10)Pour les debits consideres, lesnombres de Reynolds sont respectivement Reb ∈ 1, 3 {107; 106; 105; 104} pourla conduite Cb et Ren ∈ 3 {107; 106; 105; 104} pour la conduite Cn. Les coeffi-cients de perte de charge prennent alors les valeurs fb ∈ {16; 16; 19; 29} 10−2

et fn ∈ {19; 19; 20; 25} 10−2.


NOTATIONS

A Aire d’une section (m2)CHW Coefficient de Hazen-Williams ()DH Diametre hydraulique (m)D Diametre (m)f Coefficient de frottement ()flami Coefficient de frottement du regime laminaire ()fliss Coefficient de frottement du regime lisse ()frug Coefficient de frottement du regime rugueux ()g Gravite (m.s−2)H Charge hydraulique (m)hf Perte de charge dans une conduite (m)hs Perte de charge singuliere (m)K Coefficient de perte de charge singuliere ()Kp Conductivite hydraulique (m.s−1)L Longueur de la conduite (m)Leq Longueur equivalente (m)P Pression (Pa)Pa Pression atmospherique (Pa)Q Debit (m3.s−1)r Rugosite relative ()Re Nombre de Reynolds ()s Coordonnee curviligne (m)Sf Pente de frottement ()u Profil de vitesse (m.s−1)V Vitesse moyenne (m.s−1)Z Coordonnee verticale (m)α Constante ()ε Rugosite absolue (m)η rendement global ()ηturb rendement d’une turbine ()ηmec rendement alternateur et transformateur ()ηcond rendement des conduites ()ν Viscosite cinematique (m2.s−1)Πrec Puissance recuperable (W)ρ Masse volumique de l’eau (kg.m−3)ϕ Angle de deviation (rad)ϕ0 Angle de fermeture (rad)

Hydraulique en charge - INP Toulouse

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