Hydrostatiquepedagotech.inp-toulouse.fr/180714/res/ch1.pdf · 2021. 2. 15. · Hydrostatique...

Chapitre 1

Hydrostatique

O. Thual, 15 février 2021

Sommaire

1 Pression hydrostatique . . . . . . . . . . . . . . . . . 2

1.1 Bilan de forces . . . . . . . . . . . . . . . . . . . . . 2

1.2 Force d’Archimède . . . . . . . . . . . . . . . . . . . 3

2 Le paradoxe hydrostatique . . . . . . . . . . . . . . . 4

2.1 Basculement d’un barrage poids . . . . . . . . . . . 4

2.2 Presse hydraulique . . . . . . . . . . . . . . . . . . . 4

3 Puissance hydraulique . . . . . . . . . . . . . . . . . 6

3.1 Puissance théorique . . . . . . . . . . . . . . . . . . 6

3.2 Rendement d’une centrale hydroélectrique . . . . . . 6

1

2 Chapitre 1. Hydrostatique

Introduction

Ce chapitre est un rappel de l’hydrostatique mettant en évidence la dépendancelinéaire de la pression avec l’altitude que l’on exprime en définissant la chargehydraulique, constante dans un fluide au repos. La force d’Archimède est rap-pellée avant d’aborder le “paradoxe hydrostatique” pour lequel les forces depression peuvent résulter d’une petite force appliquée sur une petite surface(presse hydraulique) ou d’un petit volume d’eau réparti sur une grande hau-teur (exemple du barrage poids). Contrairement à certains ouvrages d’hydrau-lique, la convention qui consiste à retrancher systématiquement la pressionatmosphérique dans l’expression de la pression n’est pas adoptée ici.

1 Pression hydrostatique

1.1 Bilan de forces

Dans une cuve remplie d’eau immobile, on considère un petit volume fictifayant la forme d’un cylindre de section dA et d’axe vertical de longueur dZ(figure 1.1).

dA

dZ

�(P + dP ) dA

P dA

�g dm = dP dA

Z

Zs

Zf

Pa

Pf

P (Z)

Figure 1.1 – Bilan des forces sur une particule fluide

Les forces de pression exercées sur la surface latérale du cylindre se com-pensent. Sur la verticale, le poids du volume cylindrique d’eau est g dm où gest la gravité (g = 9, 81 m.s−2), dm = ρ dAdZ sa masse et ρ la masse volu-mique de l’eau (ρ = 103 kg.m−3). La pression P varie avec la verticale et l’ona P (Z + dZ) = P (Z) + dP . L’équilibre des forces projetées sur la verticale

Pression hydrostatique 3

s’écrit

0 = −g dm+ P dA− (P + dP ) dA = (−ρ g dZ − dP ) dA . (1.1)

On en déduit dP = −ρ g dZ, ce que l’on peut écrire dP/(ρ g) + dZ = 0. Enintégrant cette équation différentielle, on obtient

H =P (Z)

ρ g+ Z , (1.2)

où H est une constante qui ne dépend pas de l’espace. C’est la “charge hy-draulique”, exprimée dans le cas hydrostatique où la vitesse du fluide est nulle.

Sur la surface libre, de cote Z = Zs, la pression est égale à la pression atmos-phérique Pa. On peut donc écrire, pour tout point de cote Z, la relation de lapression hydrostatique

H =Paρ g

+ Zs =P

ρ g+ Z . (1.3)

On voit alors que la pression P = Pa + ρ g (Zs − Z) croit linéairement avecla profondeur Zs − Z. Par exemple, la pression au fond de la cuve, de coteZ = Zf , vaut Pf = Pa + ρ g (Zs − Zf ). Dans le cas d’une cuve possédantdes murs verticaux, la résultante des forces de pression exercées sur le fond,supposé horizontal, est égale au poids total de l’eau. Ce n’est pas le cas sil’aire de la surface libre n’est pas égale à l’aire du fond. Si l’aire de la surfacelibre est très petite, la force exercée sur le fond peut-être très supérieure aupoids de l’eau. C’est le “paradoxe hydrostatique” que l’on développe dans lesparagraphes suivants.

1.2 Force d’Archimède

“Tout corps plongé dans un liquide reçoit une force verticale égale au poidsdu volume du liquide déplacé”. On peut démontrer facilement ce principed’Archimède en considérant un corps ayant la forme d’un cylindre de sectionA et d’axe vertical de longueur Z2 − Z1 (figure 1.2). Comme la charge H =P1/(ρ g)+Z1 = P2/(ρ g)+Z2 est constante, la résultante des forces de pressionexercées sur le cylindre est une force verticale positive (vers le haut) d’intensité(P1 − P2)A = ρ g (Z2 − Z1)A = mg où m = ρ (Z2 − Z1)A est la masse duvolume d’eau qui serait contenue dans le volume Ω = (Z2−Z1)A du cylindre.On démontre également le principe d’Archimède lorsque les sections du cy-lindre ne sont pas horizontales. On généralise alors le principe au cas descorps quelconques en les découpant en petits cylindres d’axes verticaux. Laforce d’Archimède exercée sur un corps de volume Ω s’écrit donc

FArch = ρΩ g , (1.4)

où ρ est la masse volumique de l’eau.

La démonstration la plus évoluée de ce résultat, faisant appel au théorème dela divergence appliquée à des intégrales multiples, n’est pas dans l’esprit duprésent ouvrage à cause de son caractère mathématique trop avancé.


A A0

P1 A

P2 A P2 A0

Z2 � Z1

P1 A

FArch FArch

Figure 1.2 – La Force d’Archimède FArch est la résultante des forces depression.

2 Le paradoxe hydrostatique

2.1 Basculement d’un barrage poids

Pour illustrer le “paradoxe hydrostatique” par un exemple, on considère laforce de pression exercée par un volume d’eau retenu par un barrage poids(figure 1.3) pour deux configurations très différentes en terme de masse d’eaumais de hauteur d’eau h identique au contact du barrage.

Ces deux configurations génèrent exactement la même force Fpres sur le bar-rage dans la mesure où les répartitions des pressions sont identiques. Il en vade même de la force ρ g hA exercée sur le fond de surface A. Le calcul dela résultante Fpress des forces de pression exercées sur le barrage (voir exer-cice 1.1) montre que son point d’application est situé à une hauteur h/3 (centrede gravité du triangle des forces de pression) et que le barrage bascule si sonpoids Fpoids est inférieur à une valeur qui dépend de son épaisseur e et de lahauteur d’eau h.

2.2 Presse hydraulique

Un deuxième exemple illustrant le “paradoxe hydraulique” ou “principe dePascal” est celui de la presse hydraulique (figure 1.4). Une petite force f , parexemple le poids d’un cylindre de section a, induit une pression P = f/adans une cuve fermée remplie d’eau. Cette pression induit une force F = P Asur une section A, par exemple celle d’un cylindre dont le poids serait ainsicompensé. La force f induit donc une force d’autant plus grande que le rapportA/a est grand :

F = fA

a. (1.5)

En enfonçant le petit cylindre d’une profondeur dz avec une force f , on vadonc soulever le gros cylindre avec une force F = f A/a sur une hauteur dZ =dz (a/A), à cause de la conservation du volume d’eau. Le travail dW = f a dz(dont l’unité est le Joule) qui est fourni sur le petit cylindre est donc égal au

Le paradoxe hydrostatique 5

h

A

a)

b)

⇢ g h A

⇢ g h A

Fpoids

Fpoids

A

h

e

e

O

O

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Fpres

Fpres

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Figure 1.3 – Basculement d’un barrage poids sous l’effet de la pression del’eau. Les configurations a) et b) génèrent les mêmes forces.

A

Fpoids

P

P AP a

fpoids

a

Figure 1.4 – Principe de la presse hydraulique.

travail dW = F AdZ qui est reçu pour soulever le gros cylindre.


3 Puissance hydraulique

3.1 Puissance théorique

Pour prolonger et enrichir l’exemple de la presse hydraulique, on considère uneconduite reliant un tube, abritant un piston, à un réservoir dont la surface libreest située à une hauteur hbrut au-dessus (figure 1.5). On suppose que le pistonavance à une vitesse constante V : de l’énergie est récupérée en annulant sonaccélération. Si D est le diamètre du piston de section circulaire, le débit d’eauissu du réservoir est Q = AV où A = πD2/4 est l’aire de la section du piston.

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

Figure 1.5 – Force de pression sur un piston de vitesse V sans pertes decharge.

Pendant le temps dt, le travail des forces de pression exercées par l’eau d’uncôté et l’air de l’autre est égal à dW = (Pbrut − Pa)Adx avec dx = V dt. Si lapression suivait une loi hydrostatique (alors que le fluide est en mouvement),on pourrait écrire Pbrut − Pa = ρ g hbrut.Sous cette hypothèse hydrostatique, on définit la puissance théorique récupérableen empêchant le piston d’accélérer (mais pas de se mouvoir) qui est

Πtheo =dW

dt= ρ g hbrutAV = ρ g hbrutQ . (1.6)

3.2 Rendement d’une centrale hydroélectrique

En pratique, la puissance récupérable est inférieure à la puissance théoriqueΠtheo. En effet, la pression n’est pas hydrostatique à cause du mouvement dufluide et de son frottement dans les conduites qui induit des “pertes de charge”,c’est-à-dire une diminution de la pression Pnet < Pbrut au niveau du piston.On définit alors la “hauteur nette” comme étant la hauteur qui générerait

FORMULAIRE 7

la pression nette Pnet = Pa + ρ g hnet. La perte de charge dans les conduitesest alors hf = hbrut − hnet. On définit ηcond = hnet/hbrut, le rendement desconduites.

Dans une centrale hydroélectrique, le piston, présenté ici pour calculer simple-ment la puissance, est remplacé par une turbine (figure 1.6). Les frottementsdans la turbine engendrent également des pertes de charge (ηturb). Toutesles pertes de charges hydrauliques, combinées aux rendements mécaniques etélectriques de l’alternateur et du transformateur (ηmec), sont modélisées parle rendement global η, qui permet d’exprimer la puissance récupérable Πrec àtravers la relation

Πrec = η ρ g hbrutQ . (1.7)

L’ordre de grandeur de ce rendement est η= ηcond ηturb ηmec ∼ 0, 8, ce quiconduit à la relation “Πrec(kW) = 8hbrut(m)Q(m

3.s−1)” où Πrec est la puis-sance électrique récupérable exprimée en kW, hbrut la hauteur brute expriméeen m et Q le débit exprimé en m3.s−1.


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

pertes de 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

Figure 1.6 – Turbine alimentée par un réservoir. La hauteur nette hnet estcelle qui engendrerait la pression Pnet, mesurée à l’amont de la turbine, si lespertes de charge dans les conduites étaient nulles.

FORMULAIRE

Charge hydraulique hydrostatique

En l’absence de mouvement fluide, la charge hydraulique ne dépend pas del’espace et s’écrit :

H =P

ρ g+ Z .


Force d’Archimède

Un corps de volume Ω plongé dans un liquide reçoit une poussée verticale égaleau poids du volume de liquide déplacé :

FArch = ρΩ g .

Paradoxe hydrostatique ou principe de Pascal

Une petite force f exercée sur une petite section a d’un liquide confiné exerceune pression P = f/a qui induit une grande force F = P A sur une grandesection A, ce qui se traduit par la relation

F = fA

a.

Puissance récupérable

La puissance électrique récupérable Πrec à partir d’un réservoir situé à unehauteur hbrut pour un débit Q est

Πrec = η ρ g hbrutQ approximé par Πrec (kW) = 8 hbrut (m) Q (m3.s−1) ,

où le rendement η= ηcond ηturb ηmec ∼ 0, 8 prend en compte les pertes decharges hydrauliques dans les conduites et dans la turbine, ainsi que les pertesd’énergie dans l’alternateur et le transformateur.

EXERCICES

EXERCICE 1.1 Barrage poids

Un barrage poids de masse volumique ρs et de largeur l = 100 m retient unplan d’eau de profondeur h = 9 m (figure 1.3).

1) Calculer la pression P (Z) pour la cote Z ainsi que la pression P (0) aufond du plan d’eau.

2) Calculer la résultante Fpres des forces de pression exercées par l’eau surle barrage.

3) Calculer la hauteur du point d’application de cette résultante. En déduireque cette hauteur est la cote du centre de gravité du “triangle des forcesde pression” (figure 1.7).

4) En supposant que le barrage est un parallélépipède d’épaisseur e = 4 met de hauteur hb = 10 m, calculer le rapport ρs/ρ minimum des massesvolumiques, nécessaire pour éviter que le barrage ne bascule.

5) Que devient cette valeur si h = hb = 10 m ?

EXERCICES 9

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hbAAAC7HicjVLLSsNAFD2Nr1pfVZciBIvgqiRudFlw02VF+4BaapJO28E0CZOJUIuf4E50IW79EP/CP1C/wjvTFNTiY0KSM+fec2bunXEjn8fSsl4yxszs3PxCdjG3tLyyupZf36jFYSI8VvVCPxQN14mZzwNWlVz6rBEJ5gxcn9XdiyMVr18yEfMwOJXDiLUGTi/gXe45kqiTfttt5wtW0dLDnAZ2Cgql7eercwCVMP+OM3QQwkOCARgCSMI+HMT0NGHDQkRcCyPiBCGu4wzXyJE2oSxGGQ6xF/Tt0ayZsgHNlWes1R6t4tMrSGlilzQh5QnCajVTxxPtrNifvEfaU+1tSH839RoQK9En9i/dJPO/OlWLRBeHugZONUWaUdV5qUuiu6J2bn6qSpJDRJzCHYoLwp5WTvpsak2sa1e9dXT8VWcqVs29NDfB26/VqR2Me03nQtfA/n7o06C2X7Ston1M96GM8chiCzvYo1M/QAllVFClVXq4wR3ujcC4NR6Mx3GqkUk1m/gyjKcPljCapw==AAAC7HicjVLLSsNAFD2Nr1pfVZciBIvgqiRudFlw02VF+4BaSjKdtoNpEpKJUIuf4E50IW79EP/ClVv1K7wzTUEtPiYkOXPuPWfm3hk39EQsLes5Y8zMzs0vZBdzS8srq2v59Y1aHCQR41UWeEHUcJ2Ye8LnVSmkxxthxJ2B6/G6e36k4vULHsUi8E/lMOStgdPzRVcwRxJ10m+77XzBKlp6mNPATkGhtP10yWLvpRLk33GGDgIwJBiAw4ck7MFBTE8TNiyExLUwIi4iJHSc4wo50iaUxSnDIfacvj2aNVPWp7nyjLWa0SoevREpTeySJqC8iLBazdTxRDsr9ifvkfZUexvS3029BsRK9In9SzfJ/K9O1SLRxaGuQVBNoWZUdSx1SXRX1M7NT1VJcgiJU7hD8Ygw08pJn02tiXXtqreOjr/qTMWqOUtzE7z9Wp3awbjXdC50Dezvhz4NavtF2yrax3QfyhiPLLawgz069QOUUEYFVVqlh2vc4s7wjRvj3ngYpxqZVLOJL8N4/ACyXpxoAAAC7HicjVLLSsNAFD2Nr1pfVZciBIvgqiRudFlw02VF+4BaSjKdtoNpEpKJUIuf4E50IW79EP/ClVv1K7wzTUEtPiYkOXPuPWfm3hk39EQsLes5Y8zMzs0vZBdzS8srq2v59Y1aHCQR41UWeEHUcJ2Ye8LnVSmkxxthxJ2B6/G6e36k4vULHsUi8E/lMOStgdPzRVcwRxJ10m+77XzBKlp6mNPATkGhtP10yWLvpRLk33GGDgIwJBiAw4ck7MFBTE8TNiyExLUwIi4iJHSc4wo50iaUxSnDIfacvj2aNVPWp7nyjLWa0SoevREpTeySJqC8iLBazdTxRDsr9ifvkfZUexvS3029BsRK9In9SzfJ/K9O1SLRxaGuQVBNoWZUdSx1SXRX1M7NT1VJcgiJU7hD8Ygw08pJn02tiXXtqreOjr/qTMWqOUtzE7z9Wp3awbjXdC50Dezvhz4NavtF2yrax3QfyhiPLLawgz069QOUUEYFVVqlh2vc4s7wjRvj3ngYpxqZVLOJL8N4/ACyXpxoAAAC23icjVLLSgMxFD0dX7VWrWs3g0VwVWbc6FJw47KCfUAtkknTGjsvMhmhFH/AleBC/DT/QP0Kb9IpqMVHhpk5Ofeek9ybBGkoM+15LyVnaXllda28XtmoVja3tmvVdpbkiosWT8JEdQOWiVDGoqWlDkU3VYJFQSg6wfjUxDu3QmUyiS/0JBX9iI1iOZScaaKaV7W61/DscBeBX4A6ipHU3nGJARJw5IggEEMTDsGQ0dODDw8pcX1MiVOEpI0L3KFC2pyyBGUwYsf0HdGsV7AxzY1nZtWcVgnpVaR0sU+ahPIUYbOaa+O5dTbsT95T62n2NqF/UHhFxGpcE/uXbp75X52pRWOIY1uDpJpSy5jqeOGS266YnbufqtLkkBJn8IDiijC3ynmfXavJbO2mt8zGX22mYc2cF7k53n6tzuxg1ms6F7oF/vczXwTtw4bvNfxzD2XsYg8HdNhHOMEZmmiR+QD3eHRunAfnaXZbnFJxbXbwZTjPH8/WlSA=AAAC4XicjVLLSsNAFD2Nr1qrVrdugkVwVRI3uhTcuKxoH1BLSabTdmheTCZCKX6CO9GF+GH+gfoV3pmmoBYfE5KcOfeeM3PvjJ8EIlWO81KwlpZXVteK66WN8ubWdmWn3EzjTDLeYHEQy7bvpTwQEW8ooQLeTiT3Qj/gLX98puOtGy5TEUdXapLwbugNIzEQzFNEXY56fq9SdWqOGfYicHNQRT7qceUd1+gjBkOGEBwRFOEAHlJ6OnDhICGuiylxkpAwcY5blEibURanDI/YMX2HNOvkbERz7ZkaNaNVAnolKW0ckCamPElYr2abeGacNfuT99R46r1N6O/nXiGxCiNi/9LNM/+r07UoDHBiahBUU2IYXR3LXTLTFb1z+1NVihwS4jTuU1wSZkY577NtNKmpXffWM/FXk6lZPWd5boa3X6vTO5j1ms6FroH7/dAXQfOo5jo198JBEXvYxyEd9jFOcY46GmQ+xB0e8GhF1r31NLswViG/Obv4MqznD1PBl1Q=AAAC4XicjVLLSsNAFD2Nr1qrVrdugkVwVRI3uhTcuKxoH1BLSabTdmheTCZCKX6CO9GF+GH+gfoV3pmmoBYfE5KcOfeeM3PvjJ8EIlWO81KwlpZXVteK66WN8ubWdmWn3EzjTDLeYHEQy7bvpTwQEW8ooQLeTiT3Qj/gLX98puOtGy5TEUdXapLwbugNIzEQzFNEXY56fq9SdWqOGfYicHNQRT7qceUd1+gjBkOGEBwRFOEAHlJ6OnDhICGuiylxkpAwcY5blEibURanDI/YMX2HNOvkbERz7ZkaNaNVAnolKW0ckCamPElYr2abeGacNfuT99R46r1N6O/nXiGxCiNi/9LNM/+r07UoDHBiahBUU2IYXR3LXTLTFb1z+1NVihwS4jTuU1wSZkY577NtNKmpXffWM/FXk6lZPWd5boa3X6vTO5j1ms6FroH7/dAXQfOo5jo198JBEXvYxyEd9jFOcY46GmQ+xB0e8GhF1r31NLswViG/Obv4MqznD1PBl1Q=AAAC7HicjVLLSsNAFD2Nr1pfVZdugkVwVRI3uiy46bKifUAtkqTTdmiahMlEKMVPcCe6ELf+k3+gfoV3plNQi48JSc6ce8+ZuXfGT0KeSsd5yVkLi0vLK/nVwtr6xuZWcXunkcaZCFg9iMNYtHwvZSGPWF1yGbJWIpg38kPW9IenKt68ZiLlcXQhxwnrjLx+xHs88CRR54Mr/6pYcsqOHvY8cA0owYxaXHzHJbqIESDDCAwRJOEQHlJ62nDhICGugwlxghDXcYYbFEibURajDI/YIX37NGsbNqK58ky1OqBVQnoFKW0ckCamPEFYrWbreKadFfuT90R7qr2N6e8brxGxEgNi/9LNMv+rU7VI9HCia+BUU6IZVV1gXDLdFbVz+1NVkhwS4hTuUlwQDrRy1mdba1Jdu+qtp+OvOlOxah6Y3Axvv1andjDtNZ0LXQP3+6HPg8ZR2XXK7plTqlTNhchjD/s4pFM/RgVV1FCnVfq4xT0erMi6sx6tp2mqlTOaXXwZ1vMH7waYrg==AAAC7HicjVLLSsNAFD2Nr1pfVZdugkVwVRI3uiy46bKifUAtkqTTdmiahMlEKMVPcCe6ELf+k3+gfoV3plNQi48JSc6ce8+ZuXfGT0KeSsd5yVkLi0vLK/nVwtr6xuZWcXunkcaZCFg9iMNYtHwvZSGPWF1yGbJWIpg38kPW9IenKt68ZiLlcXQhxwnrjLx+xHs88CRR54Mr/6pYcsqOHvY8cA0owYxaXHzHJbqIESDDCAwRJOEQHlJ62nDhICGugwlxghDXcYYbFEibURajDI/YIX37NGsbNqK58ky1OqBVQnoFKW0ckCamPEFYrWbreKadFfuT90R7qr2N6e8brxGxEgNi/9LNMv+rU7VI9HCia+BUU6IZVV1gXDLdFbVz+1NVkhwS4hTuUlwQDrRy1mdba1Jdu+qtp+OvOlOxah6Y3Axvv1andjDtNZ0LXQP3+6HPg8ZR2XXK7plTqlTNhchjD/s4pFM/RgVV1FCnVfq4xT0erMi6sx6tp2mqlTOaXXwZ1vMH7waYrg==AAAC7HicjVLLSsNAFD2Nr1pfVZciBIvgqiRudFlw02VF+4BaSjKdtoNpEpKJUIuf4E50IW79EP/ClVv1K7wzTUEtPiYkOXPuPWfm3hk39EQsLes5Y8zMzs0vZBdzS8srq2v59Y1aHCQR41UWeEHUcJ2Ye8LnVSmkxxthxJ2B6/G6e36k4vULHsUi8E/lMOStgdPzRVcwRxJ10m+77XzBKlp6mNPATkGhtP10yWLvpRLk33GGDgIwJBiAw4ck7MFBTE8TNiyExLUwIi4iJHSc4wo50iaUxSnDIfacvj2aNVPWp7nyjLWa0SoevREpTeySJqC8iLBazdTxRDsr9ifvkfZUexvS3029BsRK9In9SzfJ/K9O1SLRxaGuQVBNoWZUdSx1SXRX1M7NT1VJcgiJU7hD8Ygw08pJn02tiXXtqreOjr/qTMWqOUtzE7z9Wp3awbjXdC50Dezvhz4NavtF2yrax3QfyhiPLLawgz069QOUUEYFVVqlh2vc4s7wjRvj3ngYpxqZVLOJL8N4/ACyXpxoAAAC7HicjVLLSsNAFD2Nr1pfVZciBIvgqiRudFlw02VF+4BaSjKdtoNpEpKJUIuf4E50IW79EP/ClVv1K7wzTUEtPiYkOXPuPWfm3hk39EQsLes5Y8zMzs0vZBdzS8srq2v59Y1aHCQR41UWeEHUcJ2Ye8LnVSmkxxthxJ2B6/G6e36k4vULHsUi8E/lMOStgdPzRVcwRxJ10m+77XzBKlp6mNPATkGhtP10yWLvpRLk33GGDgIwJBiAw4ck7MFBTE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

P (0) = Pa + ⇢ g 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

P (h) = 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