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Review Of Basic

Hydromechanics VVR090

Hydrostatics

Fluids at rest: • the at any point in a fluid is the same in every direction • in a continuous fluid with constant the pressure increases linearly with depth and the pressure is the same along horizontal planes

dp = −γ = −ρg dz

Constant density:

p − pzzh11=−γ() − =γ

1 Pressure Definitions

p : h = γ

(pressure may be expressed in height of a fluid column; e.g., mm Hg, m H2O)

p Static head (piezometric head): + z γ

p Hydrostatic pressure distribution: +=z const. γ

Force on a plane area

Total : FhA= γ c

Ic Center of pressure: yypc=+ yc A

2 Force on Curved Surfaces

Look at horizontal (Fx) and vertical components (Fz) separately.

Total force: 22 Ftot=+FF x z

Flowing Fluid

Basic equations for conservation of:

() • energy (involves potential and kinetic energy + work) • (involves momentum fluxes + )

Analysis through control

3 Classification of flow types

• 1-D, 2-D, and 3-D • real and ideal fluid • incompressible - compressible • steady – unsteady • laminar – turbulent • established – unestablished • uniform – non-uniform • subcritical – supercritical • subsonic - supersonic

Continuity Equation

1-D, steady, compressible: ρ111AVAV=ρ 2 22

1-D, steady, incompressible: A11VAV= 2 2

stream tube

4 Energy Equation

With energy losses (hL) and energy input (hM):

⎛⎞⎛⎞pV11 p 2 V 2 ⎜⎟⎜⎟++zh12 +M = ++ z ++ hL ⎝⎠⎝⎠γγ1222gg

No losses:

⎛⎞pV ⎜⎟++z = const. Bernoulli’s equation ⎝⎠γ 2g

Definitions in Fluid Flow

• Pressure head (p/g) • Elevation head (z) • Static head (piezometric head) (p/g + z) • head (V2/2g) • Total (energy) head (p/g + z + V2/2g)

• Hydraulic grade line Plane and parallell • Energy line streamlines

p +=z const. γ

5 Energy and Hydraulic Grade Lines

exit loss

flow between reservoirs

Momentum Equations

Newton’s second law (vector relationship)

∑ FQVVx =ρ()21xx −

∑ FQVVyyy=ρ()21 −

∑ FQVVzzz=ρ()21 −

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Laminar – turbulent flow Characterized by :

DVρ DV Re == μν

Critical Re-value for transition to : 2000 (pipe flow)

Head Loss in Pipes

Wall shear

L ULU22 hf== f L Dg242 Rg

A R = (hydraulic P radius)

Moody’s diagram

7 Minor Losses in Pipelines

• expansion (e.g., exit) • contraction (e.g., entrance) • bends • fittings (e.g., valves)

V 2 General expression: hK= LL2g

Pipe Configurations

Pipes in parallell:

QQQ=+++123 Q ...

hhLL====123 h L h L...

Pipes in series:

QQ===12 Q ...

hhhLL=+++123 L h L...

8 Pumps and Turbines in the System

Pumps supply the system with energy

Turbines extract energy from the system

Pumping Between Two Reservoirs

Pump and system curve

System curve:

2 HzCQP =Δ + PP

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